Self-organization on mudflats - Waddenacademie

Self-organization on mudflats Eelke Folmer

Stellingen behorende bij het proefschrift


1. “It is always advisable to perceive clearly our ignorance.” Charles Darwin (1872).

2. The fact that shorebirds attract each other and avoid interaction with conspecifics invalidates current generalized functional response models and necessitates the development of foraging models that allow for anticipation of the costs and benefits of conspecific presence. Chapters 5 and 7.

3. Future foraging models should not assume that shorebirds are able to find the best patch in terms of resource availability, but rather consider cognitive and perceptive limitations and the evolutionary origin of their current foraging behaviour. Chapter 7, inspired by McNamara and Houston (2009) Trends in Ecology and Evolution 24: 670–675.

4. Spatial autocorrelation in foraging distribution models is not only a nuisance parameter, but also a statistic that can be used to gain insight into social attraction. Chapter 3.

5. Because interference operates instantaneously, over short distances, and can be avoided by spacing out, aggregative response models based on interference costs are inadequate for the prediction of the distribution of foragers over large temporal and spatial scales. This thesis, contra Quaintenne et al. (2011) Proceedings of the Royal Society B: Biological Sciences 278: 2728-2736.

6. An unsolved problem stimulates the development of science while the persistence of a solution that is not well understood, tends to hamper it. 7. Collective scientific career building is not necessarily good for science. 8. The scientific industry tends to be preoccupied with short term pay-offs, just like banking and politics. 9. “The purpose of writing is to inflate weak ideas, obscure pure reasoning, and inhibit clarity. With a little practice, writing can be an intimidating and impenetrable fog!” Bill Waterson (1994).

10. Education towards independent and diverse thinking is a prerequisite for the development and sustainability of a fair society.

Proefschrift E. Folmer

15-03-2012

08:56

Pagina 1



15-03-2012

08:56

Pagina 2

The research presented in this thesis was carried out at the Animal Ecology Group, Centre for Ecological and Evolutionary Studies (CEES) at the University of Groningen, and at the department of Marine Ecology and Evolution of the NIOZ Royal Netherlands Institute for Sea Research. The research was financially supported by the ‘Breedtestrategie’ program of the University of Groningen. Printing of this thesis was partly funded by the University of Groningen.

Layout and figures: Dick Visser Cover design: Eelke and Hester Folmer ISBN: 978-90-367-5424-8 ISBN: 978-90-367-5445-3 (electronic version)


15-03-2012

08:56

Pagina 3

RIJKSUNIVERSITEIT GRONINGEN

Self-organization on mudflats

Proefschrift

ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. E. Sterken, in het openbaar te verdedigen op vrijdag 20 april 2012 om 16:15 uur

door

Eelke Olov Folmer geboren op 9 april 1976 te Sneek


15-03-2012

08:56

Pagina 4

Promotores:

Prof. dr. T. Piersma Prof. dr. H. Olff

Beoordelingscommissie:

Prof. dr. P.M.J. Herman Prof. dr. D.J.T. Sumpter


15-03-2012

08:56

Pagina 5

Contents

Chapter 1

Problem statement and overview

7

Chapter 2

How well do food distributions predict spatial distributions of shorebirds with different degrees of self-organization?

19

Chapter 3

The spatial distribution of flocking foragers: Disentangling the effects of food availability, interference and conspecific attraction by means of spatial autoregressive modeling

51

Chapter 4

The relative contributions of resource availability and social aggregation to foraging distributions: A spatial lag modelling approach

77

Chapter 5

Experimental evidence for cryptic interference among socially foraging shorebirds

99

Chapter 6

Seagrass - sediment feedback: an exploration using a non-recursive structural equation model

119

Chapter 7

General discussion

145

References

161

English summary

179

Nederlandse samenvatting

187

Acknowledgements

195


15-03-2012

08:56

Pagina 6


15-03-2012

1 Eelke O. Folmer

08:56

Pagina 7

Problem statement and overview


15-03-2012

08:56

Pagina 8

CHAPTER 1

Introduction

8

There is something captivating, almost supernatural, about the graceful and synchronized movements displayed by flocks of shorebirds as they fly over intertidal mudflats. Seemingly less dynamic because of its slower pace, but nevertheless just as important, is the development of seagrass beds and the balanced relationships with the physical and biotic environments. The movement patterns of flocks of shorebirds in the sky but also while foraging on mudflats, and the development of the seagrass beds, have in common that their workings are contolled by feedbacks between the elements that constitute the system. That is, through self-organization. Many systems, in both the natural world and in human society, operate or develop through self-control, i.e. without the involvement of an external regulator. This spontaneous development of a system is denoted self-organization. Globally speaking, it is the process of repeated interactions or feedbacks among elements that make up the system resulting in the spontaneous development of an element-transcending, higher level structure or function, without the intervention of an external regulator. Camazine et al. (2001) define self-organization in biological systems as follows: “a process in which a pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system. It is based on local information, without reference to the global pattern”. Self-organization occurs in a large variety of systems and is a major research topic in amongst others physics, chemistry, economics, psychology, sociology, linguistics, neuroscience and biology1 (Ball 2004). For example, a major research area in neuroscience is the role of self-organization of the physical base of memory, i.e. encoding of information in the brain through connections between neuronal assemblies (Arbib 2003). A central challenge and research theme in modern sociology and animal behaviour is to link individual decisions to group behaviour and to understand how individual decisions are influenced by the group. Over the last decades research on self-organization in biological systems has rapidly expanded (Solé & Bascompte 2006; Sumpter 2010). It is important to understand self-organization in systems because it usually is an essential inner process with substantial effects on the macro-dynamics of the system. Particularly, the stability of a system, the occurrence of catastrophes, the presence of alternate states and uncertainty are often related to feedback and propagation mechanisms in the system (Holling 1973; May 1977; Scheffer et al. 2001; Haldane & May 2011). This applies especially to complex systems, i.e. sys1

The interdisciplinary science that seeks to understand how higher level patterns result from the interactions of many elements (i.e. self-organization) is called complexity science.


15-03-2012

08:56

Pagina 9

tems whose elements interact nonlinearly. Due to their nonlinearities, complex systems may spontaneously and rapidly evolve towards unstable critical states, denoted self-organized criticality (SOC) (Bak, Tang, & Wiesenfeld 1988). Sudden, substantial changes in e.g. ecosystems such as large scale fluctuations in populations, pattern formation and extinction cascades2 are often thought to be related to SOC (Kauffman 1993; Lockwood & Lockwood 1997; Solé et al. 1999). Ecological (and many other natural and social) systems will never be completely self-organized because they will always, though to various extents, depend on one or more external variables (i.e. variables that are exogenous or independent of the system under consideration). Particularly, all the world’s ecosystems jointly constitute the biosphere which implies that all ecosystems in some way affect each other and thus depend on each other. Hence, “genuine” external variables do not exist. Nevertheless, the only realistic way to investigate the behaviour of a system is by demarcating it from other systems which can be assumed external to the system under consideration. This requires clear definitions and descriptions of the system’s components and of their interactions on time- and spatial scales, as determined by the objectives of the analysis. Particularly, on short-run time scales many variables may be considered to be independent of other variables that in the long run, however, impact on the system. From a fundamental scientific point of view, as well as for the management of ecosystems, it is important to understand the role of self-organizing processes. Particularly, it is essential to determine the degree to which system-properties are governed by external factors, and the degree to which they are the result of self-organization. Research into the behaviour of a system without considering the inner workings will not lead to adequate models to be used to analyse, predict and manage its development (Lockwood & Lockwood 2008).

Problem statement

2

Secondary extinctions that are triggered by primary extinctions.

OVERVIEW

There are many unexplored aspects of self-organization in ecology. The present study aims to contribute to the understanding self-organization by considering two different types of systems, some of their feedback mechanisms and their impacts on the higher level structures. The first is the system of foraging shorebirds on mudflats in the Dutch Wadden Sea that behave interdependently in response to other group members. The second is that of seagrasses in the Banc d’Arguin and their reciprocal relationships with the environment, particularly with the sediment characteristics of the soft-bottoms where they grow.

9


15-03-2012

08:56

Pagina 10

CHAPTER 1

A shared aim of the research in the two different systems is methodological, in that statistical models are investigated and applied to determine the strengths of feedback mechanisms. Particularly, in systems of foraging shorebirds, the level of self-organization is the outcome of the opposing forces of conspecific attraction and repulsion due to interference. The adequacy of spatial autoregression to account for, and measure, self-organization in combination with the effects of exogenous environmental factors is evaluated. Based on data collected at the intertidal mudflats of the Banc d’Arguin, the possibilities of using structural equation models are explored to determine the strength of the feedback relationship between seagrass density and sediment grain size. More specifically, in thesis project I have analysed the following topics: (1) The adequacy of classical resource-based models to predict the distributions of six species of foraging shorebirds, with varying levels of gregariousness, in the Dutch Wadden Sea at landscape level. (2) The performance of a combination of an interference-based foraging model and a conspecific attraction model to predict the distributions of foraging animals in continuous resource landscapes. This topic is addressed theoretically, by means of simulations and empirically. The model will be applied to explain flocking behaviour of the six different species of foraging shorebirds referred to under (1). (3) The adequacy of spatial autoregression to measure the impact of selforganization on flocking behaviour. This objective is investigated by means of Monte Carlo simulations. (4) To gain detailed insight into the behavioural mechanisms of interference. Since self-organization and the distributions of foraging animals in the field are the net outcomes of two opposing forces (spacing out to avoid interference and conspecific attraction to benefit from the presence of conspecifics), understanding of the working of each of the mechanisms separately requires conditions where the opposing mechanism is controlled for. In the present study the focus is on interference competition while controlling conspecific attraction. (5) To measure the strength of a feedback mechanisms between seagrass density and sediment grain size by means of a non-recursive structural equation model in the Banc d’Arguin, Mauritania.

10

Below I shall outline the 5 chapters that deal with the above objectives and make up the core of this thesis. Before doing so, I present a synopsis of classical and social foraging theory as well as of collective animal behaviour theory as introduction and framework to the first four objectives. In similar vein, a brief summary of self-organization in seagrass systems is presented as introduction to the fifth objective.


15-03-2012

08:56

Pagina 11

Theories of the spatial distribution of foragers There are several theories about the spatial distribution of foragers. In this subsection I will briefly expound why it is imperative to integrate concepts from different theories of animal behaviour for a comprehensive understanding of distributions of foraging shorebirds. The classical Ideal Free Distribution model, based on notions from optimal foraging theory, assumes that animals only suffer from the presence of conspecifics. Social foraging theory, on the other hand is based on the notion that animals may also benefit from each other. In addition, there is the theory of collective animal behavior which is relevant for the explanation of behavior of large numbers of animals. Below, I first describe the conventional optimal foraging theory and the Ideal Free Distribution theory and their limitations. Next I discuss social foraging theory and collective animal behavior theory and the way in which they complement each other. Opimal Foraging Theory and the Ideal Free Distribution Optimal foraging theory was developed to understand foraging behaviour and to predict where foragers feed and what they feed on (Emlen 1966; MacArthur & Pianka 1966). It is based on the notion that foragers are economically independent entities that behave to optimize their fitness (Stephens & Krebs 1986). Based on fitness3 optimization, Fretwell and Lucas (1969) described the equilibrium distribution of individuals across locations which they called the Ideal Free Distribution (IFD). The IFD emerges when all individuals select the most suitable location in terms of the per capita amount of resources (pay-off). Based on density-dependent suitability of the locations and the assumptions that (1) animals have perfect knowledge about the suitability of the locations (i.e. they are “ideal”) and (2) are able to freely move between, and enter, habitats at no cost (i.e. they are “free”), the IFD model makes it possible to predict the distribution of animals. The IFD model also predicts that an equilibrium will emerge where no animal can improve its pay-off by unilaterally moving elsewhere. At the equilibrium all animals experience the same pay-off. Generalized functional response models relate consumption rate to food availability and competitor density. Particularly, consumption rates depend positively on food density and negatively on the level of interference competition that foragers experience from conspecifics4 (Sutherland 1983). Interference

4

Consumption rate is often used as a proxy for fitness.

In this thesis only interference competition will be considered, because competition through depletion plays a minor role for foraging shorebirds on short time scales. Exploitative competition is distinct in that it is an indirect form of competition that operates through depletion of resources.

OVERVIEW

3

11


15-03-2012

08:56

Pagina 12

competition is defined as the reversible negative effect on intake rate through direct effects of the presence of one individual on the other (Goss-Custard 1980). When foragers are ideal and free, and the generalized functional response model is known, it is possible to derive the IFD under interference competition (van der Meer & Ens 1997)5. An important limitation of this approach is that only the costs of competition are considered while possible benefits of the presence of conspecifics are ignored6.

CHAPTER 1

Social foraging theory Basic in social foraging theory is the notion that a forager’s fitness and its behaviour depends on the behaviour of other foragers (Giraldeau & Caraco 2000; Krause & Ruxton 2002). Particularly, a forager may select its foraging location in the vicinity of other foragers to dilute the risk of being depredated (Hamilton 1971; Quinn & Cresswell 2006) or to benefit from the vigilance behaviour exercised by conspecifics (Underwood 1982). Another important reason for foraging animals to locate near other foragers, especially at the search stage, derives from the fact that foraging location decisions require information on the distribution of resources. Social foraging theory acknowledges that the presence of conspecifics may signal food availability (in addition to safety, as mentioned above). This signalling effect is especially relevant for food-searching shorebirds, because intertidal mudflats are large and their benthic prey is buried in the sediment which makes it difficult to obtain information on the distribution of benthic prey by personal sampling. Therefore, the presence and behaviour of other foragers are informative in that they signal the presence of resources (and the absence of danger). Following or joining other foragers may thus be beneficial in that search costs and predation risk are reduced. Particularly, a group of foragers may synchronize their behaviour via behavioural feedbacks to decrease predation costs and to increase foraging opportunities. To sum up, through conspecific interaction, animals can enhance their ability to detect resources and danger in the environment. Particularly, interactions with others allow individuals to evade their own cognitive and perceptive limitations resulting in more accurate and faster decision-making.

12

5 Interference competition can be reduced by spacing out. It should be noted that such costreducing behaviour is not included in interference models. Instead, animals are modelled as moving like “aimless billiard balls” (van der Meer & Ens 1997) . These authors noted further that the absence of possible avoidance of interference in interference models is inconsistent from an optimization point of view. It should be pointed out that interference models are also unrealistic in the sense that the size of patches is not explicitly considered and that interactions between animals are simply functions of forager density. 6

It is interesting to note that Fretwell and Lucas explicitly point out this limitation in their original paper.


15-03-2012

08:56

NET COST

Pagina 13

NET GAIN

ZERO

benefit of proximity

+

benefit

net

0 cost of proximity

–

(LOW)

inter-individual distance

(HIGH)

Figure 1.1. Cost and benefit of conspecific proximity as functions of distance between foraging animals. The cost of interference is large on short distances but rapidly decline when interindividual distances increase. The benefit of sociality is large at short and intermediate distance and levels off to 0 when inter-individual distances increase. The net benefit curve is hump-shaped with the maximum net benefit at intermediate distances.

Locating in the vicinity of conspecifics, however, may lead to increased interference competition. Hence, there is a distance-dependent trade-off between benefits and costs of locating in the vicinity of conspecifics (Figure 1.1). For shorebirds foraging on large open intertidal mudflats the benefits associated with the presence of conspecifics are likely to outweigh the interference costs because food patches tend to be large such that the costs due to interference competition can easily be reduced by spacing out.

7 Collective grouping and movement patterns are not typical for shorebirds but are also observed in insects, fish, mammals and other species of birds.

OVERVIEW

Collective animal behaviour To understand the foraging behaviour of a flock of shorebirds, not only the behaviour of the individuals needs to be understood but also how they act together to form the behaviour of the flock. As pointed out by Gell-Mann (1997) in a different context: “It is vitally important that we supplement our specialized studies with serious attempts to take a crude look at the whole.” Models of collective animal behaviour are based on the notion that patterns may arise in large groups of similar individuals through repeated interactions (Sumpter 2010). However, because the flock cannot be described without describing the behaviour of individual foragers, and because the behaviour of individuals must be described with reference to the behaviour of conspecifics, the shaping of flocks is difficult to model7.

13


15-03-2012

08:56

Pagina 14

Understanding the behaviour of large assemblies of individuals has mainly been advanced by mathematical models, individual-based simulations, or a combination of both. Regarding flocking behaviour, simulation is an appealing approach because it proceeds on the basis of simple assumptions about how individuals behave and respond to each other. Computer simulations with multiple agents programmed with simple neighbourhood rules like: “when far away, move towards and when too close, move away” has put forward simple and compelling explanations of seemingly choreographed phenomena (Couzin & Krause 2003; Sumpter 2010). Another advantage of simulation is that it can straightforwardly produce empirically testable hypotheses on patterns of collective behaviour. For these reasons, simulation will be applied below to gain insight into the performance of the food availability-conspecific attractioninterference competition model to predict the emergence of flocks of foraging animals and their spatial distribution.

Self-organization of seagrass systems

CHAPTER 1

Nontrophic interactions that modify abiotic environments and shape communities are common features of ecosystems. In this context ecosystem engineers, i.e. species that fundamentally impact on their abiotic environment8 (Jones, Lawton, & Shachak 1994; Wright & Jones 2006) play a crucial role. Ecosystem engineers may impact on the development of spatial structure in the environment and affect the availability of resources to other species (Rietkerk et al. 2004). However, there may also be reverse effects: the abiotic environment may affect the population dynamics of the engineer (Cuddington, Wilson, & Hastings 2009). Complexity theory predicts that ecosystems that develop through self-organization may become resilient to change, but may also suddenly shift to alternative states (Levin 1998; Scheffer and Carpenter 2003). It is important to understand the mechanisms of self-organization of ecosystem engineers in relation to exogenous factors because these factors jointly impact on the trajectory and stability of the system. Particularly, the reinforcing feedback between the engineer and its environment may result in the development of a channeled, ordered9 and resilient ecosystem that in response to changing environmental

14

8 The definition of “ecosystem engineer” is not strict because all species modify their abiotic environment in some way or another on varying temporal and spatial scales. 9

Kauffman calls self-organizing systems “anti-chaotic” because, despite different initial conditions, the same final state may develop


15-03-2012

08:56

Pagina 15

conditions10, may be subject to catastrophic shifts (Kauffman 1993; Levin 1998; Scheffer & Carpenter 2003). Seagrasses growing on intertidal mudflats are ecosystem engineers in that they affect the hydrology and sediment properties. Seagrass may locally promote its own growth in that the canopy reduces mechanical disturbance by reducing water flow velocities. The reduced water flow velocities also reduce erosion and stimulate deposition of fine sediments and associated nutrients. In turn, the accumulated fine sediments influence seagrass growth. Particularly, in silty and anoxic sediments, high concentrations of organic matter also result in the production of hydrogen sulfide that, depending on the concentration, may negatively affect seagrass growth. Detailed understanding of the feedback mechanism between seagrass density and sediment properties is critical to predict the responses of a seagrass-dominated ecosystem to environmental change. Particularly, it is important to know the strength of the feedback between seagrass growth and the environment in relation to external factors.

Outline of the study This thesis is made up of two parts. The first deals with self-organization of foraging shorebirds, the second with self-organization in seagrass beds. Below, I first outline the foraging shorebird chapters, next the seagrass chapter.

10

It should be observed that such a system cannot be too sensitive for else it could not have evolved to its present state in the first place.

OVERVIEW

Part A: Self-organization of foraging shorebirds This main purpose of this part of the study is to increase understanding of the distribution of foraging shorebirds by addressing research questions 1 - 4. The research questions are addressed conceptually, theoretically, by means of simulation, via observational studies and statistical modeling, and by indoor aviary experiments. To understand distributions of foraging shorebirds I consider foraging flocks of shorebirds as assemblies of interconnected individuals responding to their exogenous environment and to conspecifics. As noted above, there are positive and negative sides to the presence of conspecifics. The net benefit may be optimized by means of spacing out in such a way that interference is avoided while the benefits of conspecific presence are still gained. Chapter 2 analyses the distribution of six different species of foraging shorebirds in the Dutch Wadden Sea at landscape level on the basis of a resource

15

CHAPTER 1


16

15-03-2012

08:56

Pagina 16

based model. The main hypothesis analysed is that resource-based models have better predictive power for interference-sensitive species than for interferenceinsensitive species because the former are less influenced by conspecific attraction and thus will be more driven by resource availability than the latter. Another hypothesis that will be tested is that species that are interference-insensitive will be more clustered than predicted by the spatial distribution of their food resources because they are more responsive to conspecific attraction such that they follow each other when selecting foraging patches. Chapter 3 develops a theoretical framework of foraging distributions of gregarious animals in continuous resource landscapes. A classical interferencebased foraging model is combined with a conspecific attraction model which is used to simulate distributions of foragers in continuous resource landscapes. In this model the cost of interference and benefits of the presence of conspecifics depend on inter-individual distances. Particularly, it is assumed that interference may be mitigated by maintaining short distances to conspecifics and that the benefits of conspecific presence (causing attraction) operate over larger distances (see Figure 1.1). Analysis of conspecific attraction has been hampered by the lack of an operational definition and adequate measurement methods. This chapter proposes spatial autoregression to measure self-organization based on the assumption that behavioural feedback amongst shorebirds manifests itself as spatial dependence (i.e. the tendency of foragers to choose locations in the vicinity of other foragers). To account for the fact that animals copy each other’s behaviour, the spatial multiplier is proposed to measure the total food effect. Uncertainty in the forager’s knowledge about the food distribution and consequently in the spatial distribution of the foragers are included the model. The theoretical model and the adequacy of autoregression are tested by means of numerical simulations. In Chapter 4 the impact of self-organization in relation to the effects of exogenous factors (i.e. food availability and abiotic habitat characteristics) on the distribution of six species of shorebirds in the Dutch Wadden Sea are considered. The operational definition of conspecific attraction and spatial autoregression introduced in Chapter 3 are applied here. In this chapter the scale of investigation is much smaller and is more behaviourally oriented. The model is estimated on different spatial resolutions to get insight into the modifiable areal unit problem (MAUP), i.e. the problem that regression estimates change by level of aggregation. Lastly, the spatial multiplier is applied to obtain the total food effect. To evaluate the appropriateness of generalized functional response functions, Chapter 5 investigates the costs and the underlying mechanisms of interference competition by means of experiments with red knots. As noted above, animals


15-03-2012

08:56

Pagina 17

anticipate the presence of conspecifics and may try to avoid physical interactions. In a novel experimental setup with a small moving patch, the behaviour of red knots is analysed to unravel the foundation of interference. Because there may be important differences between individuals of shorebirds, the dominance status is explicitly considered.

OVERVIEW

Part B: Self-organization in seagrass systems In Chapter 6 the presence and density of seagrass on the intertidal flats of the Banc d’Arguin and their reciprocal relationships with sediment characteristics is analyzed. The overall objective of the chapter is to contribute to the understanding of the functioning of soft-bottom intertidal seagrass ecosystems. The strength of the feedback mechanisms are estimated by means of a non-recursive structural equation model (SEM). Chapter 7 synthesizes the findings of the research and discusses possibilities for future research.

17


15-03-2012

08:56

Pagina 18


2

15-03-2012

08:56

Pagina 19

How well do food distributions predict spatial distributions of shorebirds with different degrees of self-organization?

Eelke O. Folmer, Han Olff and Theunis Piersma

Abstract Habitat selection models usually assume that the spatial distributions of animals depend positively on the distributions of resources and negatively on interference. However, the presence of conspecifics at a given location also signals safety and the availability of resources. This may induce followers to select contiguous patches and causes animals to cluster. Resource availability, interference and attraction therefore jointly lead to self-organised patterns in foraging animals. We analyse the distribution of foraging shorebirds at landscape level on the basis of a resource-based model to establish, albeit indirectly, the importance of conspecific attraction and interference. At 23 intertidal sites with a mean area of 170 ha spread out over the Dutch Wadden Sea, the spatial distribution of six abundant shorebird species was determined. The location of individuals and groups were mapped using a simple method based on projective geometry, enabling fast mapping of low tide foraging shorebird distributions. We analysed the suitability of these 23 sites in terms of food availability and travel distances to high tide roosts. We introduce an interference sensitivity scale which maps interference as a function of inter-individual distance. We thus obtain interference-insensitive species which are only sensitive to interference at short inter-individual distances (and may thus pack densely) and interference- sensitive species which interfere over greater inter-individual distances (and thus form sparse flocks). We found that interference-insensitive species like red knot (Calidris canutus) and dunlins (Calidris alpina) are more clustered than predicted by the spatial distribution of their food resources. This suggests that these species follow each other when selecting foraging patches. In contrast, curlew (Numenius arquata) and grey plover (Pluvialis squatarola), known to be sensitive to interference, form sparse flocks. Hence, resource-based models have better predictive power for interference-sensitive species than for interferenceinsensitive species. It follows from our analysis that monitoring programmes, habitat selection models and statistical analyses should also consider the mechanisms of self-organization.

Journal of Animal Ecology 79:747-756.


15-03-2012

08:56

Pagina 20

CHAPTER 2

Introduction

20

In the theoretical resource-based literature, animal-habitat relationships are derived from fitness or intake maximization (Fretwell & Lucas 1969; Kacelnik, Krebs & Bernstein 1992). Intake rates are assumed to depend on resource density and interference (Beddington 1975; Ruxton, Gurney & De Roos 1992; Moody & Houston 1995). Under the assumption that animals behave ideally and freely and maximise intake rates, aggregative response functions may be derived (Sutherland 1983; Moody & Houston 1995; Van der Meer & Ens 1997). This approach bases predictions of the spatial distribution of foraging animals on straightforward mechanistic principles. The empirical resource-based literature takes a phenomenological approach and investigates relationships between habitat characteristics and animal densities statistically (Bryant 1979; Piersma et al. 1993; Yates et al. 1993; Zwarts, Wanink & Ens 1996; Guisan & Zimmermann 2000; Manly et al. 2002; Granadeiro et al. 2007). These studies find mixed results and heterogeneous relationships, amongst others because animal densities may depend on habit characteristics in non-linear ways. Specifically, ecological factors may impose upper or lower limits on response variables so that the impacts within and outside the limits substantially differ (Thomson et al. 1996; Cade & Noon 2003). Put differently, ecological factors may operate as constraints on, rather than as exact determinants of behaviour. Moreover, multiple limiting factors may interact. Both the theoretical and the empirical literature are pre-occupied with the negative impacts of co-occurrence of conspecifics while possible benefits are frequently ignored. The presence of many eyes and ears in a group increases the chance that predators (Pulliam 1973; Beauchamp 1998; Krause & Ruxton 2002; Whitfield 2003) or resources (Valone & Templeton 2002; Danchin et al. 2004) are detected. Additionally, animals may have developed social behaviour in response to past selection pressures (Byers 1997). Behaviour of individuals thus depends on behaviour of group-members (Sirot 2006). Therefore not only negative aspects of interdependent relationships among individuals must be considered when studying habitat selection but also the positive aspects (Melles et al. 2009). It follows that the gregarious nature of animals may be a source of heterogeneity in the relationship between foraging animals and habitat characteristics. Conspecific attraction is not necessarily beneficial, but may also lead to the selection of suboptimal foraging patches (Giraldeau, Valone & Templeton 2002). Specifically, if predecessors choose suboptimal foraging patches and followers copy their behaviour and select contiguous or nearby patches, a collective mistake results. Hence, conspecific attraction may lead to a mismatch

15-03-2012

08:56

Pagina 21

between the spatial distribution of foraging animals and the spatial distribution of their food. For solitary species the risk of mismatch is smaller because patch selection will be based on expected intake rate only rather than on a combination of expected intake rate and conspecific attraction. Recently, the concept of self-organisation has been introduced to understand collective animal behaviour of groups without permanent leaders (Camazine et al. 2001; Krause & Ruxton 2002; Sumpter 2006). Central to this line of work is the notion that group formation results from repeated interactions among neighbours. These types of models view animals as interacting particles that make movement decisions in response to the locations and movements of their neighbours (Reynolds 1987; Couzin et al. 2002). This framework has been useful to understand and predict properties of groups with many individuals, such as insect swarms, fish schools and bird flocks (Sumpter 2006). Flocks of shorebirds, particularly of dunlin (Calidris alpina) and red knot (Calidris canutus), may consist of many thousands of individuals displaying synchronized movements in flight, often in response to predation (Piersma et al. 1993; Van de Kam et al. 2004; Van den Hout, Spaans & Piersma 2008). These flocking patterns are maintained during foraging (Goss-Custard 1970). Despite their ubiquity, flocking patterns tend to be ignored in most studies of low-tide shorebird spatial distributions, as it is generally assumed that animal-habitat relationships only result from individual choices in response to resources and interference (Nehls & Tiedemann 1993; Piersma et al. 1993; Van Gils & Piersma 2004; Vahl et al. 2005; Van Gils et al. 2006; Spruzen, Richardson & Woehler 2008). We hypothesize that shorebirds choose foraging patches based on exogenous factors (e.g. food availability, danger and travel costs) and, at varying degrees, in response to the presence of conspecifics (Fig. 2.1). Handling time and prey type determine the distance between conspecifics. For instance, oystercatchers (Haematopus ostralegus) foraging on bivalves, require long handling times making it possible for competitors to steal prey (kleptoparasitism) (Ens, Esselink & Zwarts 1990; Stillman et al. 2002). Hence, oystercatchers are sensitive to interference and therefore maintain relatively large inter-individual distances (Moody et al. 1997). In contrast, for species with short handling times (e.g. red knot) the cost of interference is small and animals may easily form dense flocks (Van Gils & Piersma 2004). Hence, they are interference-insensitive, i.e. there is a small impact of interference on spacing behaviour. The objective of this paper is to test the hypothesis that resource availability and distance to high tide roost are more important determinants of the spatial distributions of interference-sensitive species than of interference-insensitive species. This will be reflected in a larger residual variance of a regression of bird density on these predictors for the latter than for the former. The reason is that

SELF-ORGANIZATION AND LANDSCAPE-LEVEL SITE-CHOICE


21


15-03-2012

08:56

Pagina 22

attraction

weak interference

no attraction

2

3

4

strong interference

1

CHAPTER 2

Figure 2.1. Clustering of foraging shorebirds as a function of food availability, attraction and interference. The darker the patch color, the higher the food availability. 1: weak interference and no attraction: clustering at the patch with the highest food density; 2: weak interference and strong attraction: strong clustering at the patch with the highest food density; 3: strong interference and no attraction: weak-to-moderate clustering at the patch with the highest food density and weak clustering at patches with low food densities; 4: strong interference and attraction: moderate clustering at the patch with the the highest food density and weak clustering at the patch with the next highest food density.

22

in the case of interference-insensitive species systematic predictors (i.e. the joint impact of conspecific attraction and interference-insensitivity) are missing. The hypothesis will be tested, at landscape scale, for six common shorebird species in the Dutch Wadden Sea. The Wadden Sea is an area par excellence to study resource availability–animal density relationships. First, many shorebird species in the Wadden Sea are abundant (Zwarts & Wanink 1993; Van de Kam et al. 2004). By focussing on abundant species, the role of accidental relationships (i.e. relationships that may occur by chance) is reduced. Secondly, there is detailed information available on food availability in the Wadden Sea because of an ongoing benthos monitoring programme (Piersma et al. 1993, 1995; Van Gils et al. 2007; Kraan et al. 2009a). Thirdly, there is large variation in food density and in the level of flocking between shorebird species (Goss-Custard 1970). Finally, the Wadden Sea is an open and well-known landscape such that the risk of not identifying possible confounding site characteristics affecting dispersion is small. Moreover, even if they are overlooked, they may not affect the analysis when they are constant between species.


15-03-2012

08:56

Pagina 23

Methodology

Benthos sampling As part of a long-term benthic research programme (Piersma et al. 1993; Kraan et al. 2009a,b), we determined the density of macrozoobenthos in the Dutch Wadden Sea between July and September 2004. Benthos sampling was performed over 250 m grids (Fig. 2.2). The sampling stations were visited by foot during low tide and by boat during high tide (by boat to maximally utilize the number of working hours while in the field). When sampling by foot, one sample was taken at each station. Each sample consisted of sediment taken down to a depth of 20–25 cm with a core with area of 1/56 m2. The top (0–4 cm) layer of the sample was separated from the bottom layer. The top and bottom layers were sieved separately over 1-mm mesh. Since polychaetes are able to move from the bottom to the top part layer, their vertical location in the layer was not recorded. At the same locations, mudsnails (Hydrobia ulvae) were also sampled but with a smaller core (1/267 m2) to a depth of 4 cm. Mudsnail samples were sieved over a 0.5 mm mesh. When sampling by boat, at each station two samples were taken, down to a depth of 2025 cm, each with a core with area of 1/115 m2. We took two samples to obtain similar precision of benthos density estimates as in the samples collected by foot. The two samples were sieved jointly. Due to practical limitations, for these samples the top layers were not separated from the bottom layer. In the field the numbers of adult and juvenile individuals of each macrobenthos species were counted. All molluscs and shore crabs (Carcinus maenas) that were retained in the sieve were frozen at –20 ºC for later analysis in the laboratory. In the laboratory the lengths of all individual specimens were measured to the nearest 0.1 mm. For bivalves, the flesh was separated from the shell and dried at 55–60 ºC. After determination of the dry mass (to the nearest 0.1 mg), the flesh was incinerated at 550 ºC for 2 hours. The weights of the ashes were measured to the nearest 0.1 mg. In this way species and length specific values


The study area The Dutch Wadden Sea is shallow and contains large soft-sediment flats that emerge twice a day during low tide. The mudflats alternate with permanent channels (Fig. 2.2). The flats are characterized by smooth gradients both in terms of physical properties, such as sediment grain size distributions, and biological properties, such as density of macro-zoobenthic species (Kraan et al. 2009a). Due to the semidiurnal tides, the mudflats are accessible to shorebirds approximately twice per day. High tide roosts of non-breeding shorebirds are found on the mainland and on all islands (Koffijberg 2003; Van de Kam et al. 2004).

23

24 0

mudflat

15

30 km

08:56

censused sites

benthos sample stations

15-03-2012

Figure 2.2. Map of the Dutch Wadden Sea. Benthos sampling stations and the sites for which shorebird distributions were mapped are indicated.

CHAPTER 2

Proefschrift E. Folmer Pagina 24


15-03-2012

08:56

Pagina 25

Bird mapping For the 23 sites that were sampled for benthos, shorebird distributions were mapped (Fig. 2.2). The maps were drawn between three days before and three days after benthos sampling. Several studies (e.g. Piersma et al. (1993) and Van Gils et al. (2003)), show that depletion and death of benthic species affecting their densities occur over longer time periods than six days. Benthos sampling and shorebird mapping took place around the centres of the mudflats where submersion times are shortest. This ensures accessibility of the mudflats for most of the time throughout the tide. Observation points were chosen centrally on the mudflats (>1 km away from gullies). Bird distributions were mapped in between two hours before until two hours after low tide. The area of exposed mudflat changes little in this time span so that the spatial distribution of the birds is not affected by tidal movement. Furthermore, disturbance due to the presence of the observer is minimal under these conditions, because the extent of available mudflat is at its largest. The observer (EOF) arrived at the observation points by foot well before mapping started, so that disturbed birds would have sufficient time to return to the areas through which the observer had arrived. Observations were started in opposite direction from which the observer had arrived. Only Curlew (Numenius arquata) seemed disturbed and was never recorded within 200 m from the observer. Positions of individuals and flocks were determined with the aid of GPS, compass and rifle scope with a ranging reticle (mill dots). GPS was used to determine the position of the observer; the compass to determine the observation direction. The rifle scope mounted on the telescope enabled the observer to measure the distance between each individual bird or flock edges and the horizon in terms of mill dots. Based on principles of projective geometry this distance was used to calculate the true distance from the observer (Heinemann


for ash-free dry mass (AFDM) were obtained. Further details about prey sampling and analysis can be found in (Piersma et al. 1993, 1995, 2003; Kraan et al. 2009a). For the specimens counted in the field and not brought to the lab (polychaetes and isopods), we obtained estimates of energy values from the literature (Appendix S1). Note that the values thus obtained are approximations. This, however, is not a problem in the present analysis, since its objective is to analyse the significance of food as predictor of patch choice and flocking variance rather than precisely estimating and comparing regression coefficients of food variables. Particularly, some inaccuracy in the regression coefficients does not affect the predictive power of the estimated models for the entertained objectives of the paper. Moreover, we also considered higher than conventional (5%) significance levels of the coefficients.

25


15-03-2012

08:56

Pagina 26

CHAPTER 2

1981). The procedure was regularly calibrated using objects with known locations and true distances. All individual birds and flocks in a 360° circle around the observer were plotted on maps with 100 m grids. Individual birds were plotted as points and flocks as polygons in which the numbers of individuals were registered. In some cases flocks rather than individuals were considered because when the number of birds covering a small area was large it was not feasible to plot all individual positions. In early morning and late afternoon visibility could be poor due to reflecting light making it impossible to make a full 360° map. In those cases observations in the direction of poor visibility were cancelled. During a single low tide period, depending on the average bird density, either one or two censuses were done on different sites on the same mudflat. A typical census would relate to a circular area with radius between 650 and 800 m. This census area, denoted “site”, is the spatial unit of analysis below.

26

Data preparation Regarding benthos availability for short-billed birds potentially feeding on small bivalves (i.e. red knot and dunlin), we only considered bivalves in the top-layer (Van Gils et al. 2009). Since it was not possible to separate the top and bottom layers for samples collected by boat, we obtained estimates for the top layer benthos in this case by using the proportion of top layer benthos found in samples collected by foot. The proportions of benthos in the top and bottom layers may differ between species, size classes and between the eastern and western Wadden Sea (Van Gils et al. 2009). We therefore used species-, size- and location (western (A-C) and eastern (D-L) Wadden Sea (see Fig. 2.2)) specific proportions. For example, if the proportion of top layer ingestible Baltic tellin (Macoma balthica, 10 mm (1)

2 (Goss-Custard 1996)

compilation

Macoma

1,2,3

>10 mm

3 (Ens et al. 1996)

Mytilus

1,2

>12

Schiermonnikoog, DWS

Scrobicularia

1,2

>10 mm (1)

Nereis

1,2,3

50 mm (0.086 A)

1 (Zwarts et al. 1996)

(1) A (Zwarts & Wanink 1993)

CHAPTER 2

DWS: Dutch Wadden Sea

44

sampling location Frisian Coast, DWS


15-03-2012

08:56

Pagina 45

3

oyc 0 17.84

ner 0.00 39.49

cer 0.00 3.63

mac 0.00 6.54

myt 0.00 0.95

scr 0.00 7.3

dhtr. 0.0 4.9

silt 0.0 72

elev. -72

Figure S1.4. Mean density of oystercatcher (individual ha-1), its prey species (g. AFDM m-2) and physical characteristics for 23 sites. Abbreviations: cer: Cerastoderma edule, mac: Macoma balthica, myt: Mytilus edulis, ner: Nereis diversicolor, scr: Scrobicularia plana, dhtr.: distance to high tide roost (km), silt: silt content (%), elev.: elevation of mudflat (cm +NAP).


H A2 A1 G2 G1 I2 D1 A3 J1 K2 F1 K1 F2 D2 L A4 I1 B C1 E1 E2 J2 C2

45


15-03-2012

08:56

Pagina 46

Curlew (Numenius arquata)

Table S1.5. Summer prey of curlew identified in the literature. The numbers in the column “references” denote the literature (given in the right hand table) in which the prey under consideration has been found to be part of the diet. If the prey species forms a significant part of the diet, the reference numbers are bold. The column “size class and/or AFDM” gives the mean estimated length for each species or/and approximate energy content (mg AFDM) of an individual as used in our analyses. The given value of energy contents are accompanied with a capital letter corresponding to the source of this information (given in the right hand table). For some species AFDMs could not be identified. In those cases AFDMs of individuals were guessed on the basis of comparison to similarly sized species. If a prey species was found to be reasonably abundant in the Wadden Sea its name is in bold; it is included in the full regression model. prey

references

size class and/or AFDM (g)

source

sampling location

1 (Goss-Custard et al. 1977)

The Wash, UK

Carcinus

1,2,3,4

0.035*

2 (Petersen & Exo 1999)

Spiekeroog, GWS

Crangon

2,3

25 mm (0.200 B)

3 (Ens, Esselink, & Zwarts 1990) Frisian Coast, DWS

Arenicola

1,3

0.15 (guess)

4 (Goss-Custard & Jones 1976)

Cerastoderma

1

small (1,3)

A (Zwarts & Wanink 1993)

Coast of Frl. DWS

Scrobicularia

1

small (1,3)

B (Nehls & Tiedemann 1993)

Königshafen , GWS

Mya

3

small (1,3)

Macoma

1

small (1,3)

Lanice

1,2,4

0.07 (guess)

Nereis

1,2,3

50 mm (0.086 A)

DWS: Dutch Wadden Sea GWS: German Wadden Sea

The Wash, UK

Frl.: Friesland

* mean of east and west Few small mya

CHAPTER 2

Notes Goss-Custard et al. found that curlews only select Carcinus smaller than 35 mm (carapace width). The largest Carcinus in our samples was 20 mm.

46


15-03-2012

08:56

Pagina 47

2

curlew 0 16.69

ner 0.00 6.24

are 0.00 26.41

lan 0.00 7.2

cra 0.0 2.29

car 0.00 7.3

dhtr. 0.0 4.9

silt 0.0 72

elev. -72

Figure S1.5. Mean density of curlew (individual ha-1), its prey species (g. AFDM m-2) and physical characteristics for 23 sites. Abbreviations: are: Arenicola marina, car: Carcinus maenas, cra: Crangon crangon, lan: Lanice conchilega, ner: Nereis diversicolor, dhtr.: distance to high tide roost (km), silt: silt content (%), elev.: elevation of mudflat (cm +NAP).


G2 J1 K1 E1 J2 I2 D1 I1 H G1 L E2 A4 F1 K2 D2 A3 A1 F2 C2 C1 A2 B

47


15-03-2012

08:56

Pagina 48

Grey plover (Pluvialis squatarola) Grey plover is a visual hunter; it mainly hunts for large worms on mudflats that contain less than 5% silt. None of the sites that were censused contained more than 5% silt.

Table S1.6. Summer prey species of dunlin identified in the literature. The numbers in the column “references” denote literature (given in the right hand table) in which the prey under consideration has been found to be part of the diet. If the prey species forms a significant part of the diet, the reference numbers are bold. The column “size class and/or AFDM” gives (1) for bivalve species: the ranges of the lengths that are profitable and ingestible (numbers within parentheses refer to references in the right-hand table) (2) for polychaetes and crustacea: the estimated mean length per species or/and approximate energy content (mg AFDM) of an individual as used in our analyses. The given value of approximate energy contents are accompanied with a capital letter corresponding to the source of this information (right hand table). For some species AFDMs could not be identified. In those cases AFDMs of individuals were guessed on the basis of comparison to similarly sized species. If a prey species was found to be reasonably abundant in the Wadden Sea its name is in bold; it is included in the full regression model. prey

references

size class and/or AFDM (g)

source

sampling location

Nereis

1,2,3,4

50 mm (0.086 A)

in (Smit & Wolff 1980)

Ameland, DWS

Scoloplos

2

0.01

2 (Goss-Custard et al. 1977)

The Wash, UK

1 Kersten and Piersma

Nephthys

(guess)

60 mm (0.031 A)

3 (Esselink & van Belkum 1986) Dollard, DWS

Heteromastus

6

0.005 (guess)

4 (Ruiters 1992)

Westerschelde, DD

Lanice

2

0.07

(guess)

5 (Pienkowski 1983)

(guess)

Holy Island Sands, Northumberland, UK

Arenicola

5

0.15

Macoma

2

small

6 (Kersten & Piersma)

Ameland, DWS

Cerastoderma

2

small

A (Zwarts & Wanink 1993)

Coast of Frl. DWS

Carcinus

2

0.035*

Corophium

2,5

6 mm (0.001 A)

DWS: Dutch Wadden Sea DD: Dutch Delta UK United Kingdom

* mean of west and east

CHAPTER 2

Notes We did not find evidence in the literature that Nephthys has been found to be part of the diet of grey plover. We consider it an unlikely a priori hypothesis that grey plover would ignore Nephthys; we therefore included it in the full model.

48


15-03-2012

08:56

Pagina 49

1

plover 0 20.17

ner 0.00 3.05

sco

0.00 1.48

nep

0.00 5.05

het lan

0.00 42.13

are

0.00 5.03 0.00

cor dhtr. silt

0.09 0.00 7.3 0.0 4.9 0.0

elev.

72 -72

Figure S1.56. Mean density of grey plover (individual ha-1), its prey species (g. AFDM m-2) and physical characteristics for 23 sites. Abbreviations: are: Arenicola marina, cor: Corophium volutator, het: Heteromastus filiformis, lan: Lanice conchilega, ner: Nereis diversicolor, nep: Nephthys hombergii, sco: Scoloplos armiger, dhtr.: distance to high tide roost (km), silt: silt content (%), elev.: elevation of mudflat (cm +NAP).


K1 G2 I2 H A1 G1 J1 K2 J2 A4 I1 L F1 E1 E2 C2 C1 A3 A2 B D1 D2 F2

49


15-03-2012

08:56

Pagina 50


3

15-03-2012

08:56

Pagina 51

The spatial distribution of flocking foragers: Disentangling the effects of food availability, interference and conspecific attraction by means of spatial autoregressive modeling

Eelke O. Folmer, Han Olff and Theunis Piersma

Abstract Patch choice of foraging animals is typically assumed to depend positively on food availability and negatively on interference while benefits of the co-occurrence of conspecifics tend to be ignored. In this paper we integrate a classical functional response model based on resource availability and interference with a conspecific attraction model and use it to simulate spatial distributions of animals in their continuous resource landscapes. We consider both equilibrium and non-equilibrium distributions. We show that the integrated model produces distributions of foraging animals that closely match the distributions observed in nature. The simulations also show that under information uncertainty the locations of flocks are highly variable when conspecific attraction is strong. We furthermore explain how we can estimate the impacts of conspecific attraction and interference on the distribution of foraging animals by spatial autoregression. On the basis of simulated data we show that the separate impacts of interference and conspecific attraction can be disentangled when prior information on either is available, in addition to information on resource density and predator density, and that the total food effect is given by the spatial multiplier.

Oikos 2011


15-03-2012

08:56

Pagina 52

CHAPTER 3

Introduction

52

The understanding of the spatial distribution of foraging animals is one of the most important themes in theoretical and empirical ecological research. The “Ideal Free Distribution” (IFD) model (Fretwell and Lucas 1969) describes the equilibrium distribution of individuals over resource patches based on fitness maximization. When combined with a generalized functional response model that relates food consumption to food availability and competitor density, the IFD model allows prediction of the spatial distribution of foraging animals. The basic theoretical result of this combined model, that we denote the “classical model”, is that the degree of aggregation in patchy resource landscapes depends positively on food availability and negatively on interference (Sutherland 1983, Sutherland and Parker 1992, Moody and Houston 1995, van der Meer and Ens 1997). A shortcoming of the classical approach (besides the unrealistic assumptions that animals are omniscient, have equal competitive abilities, and incur no cost of moving), is that beneficial effects of the presence of conspecifics are ignored (Muller et al. 1997). Particularly, the presence of conspecifics dilutes predation risk (Hamilton 1971, Pulliam 1973, Quinn and Cresswell 2006) and it signals the availability of food and safety which reduces search costs, and costs related to exercising vigilance while foraging (Underwood 1982). Hence, the predictive and explanatory power of aggregative response models that do not take into account these self-organizing effects, may be poor for many species (Amano et al. 2006, Folmer et al. 2010). The role of signalling has been explored in the animal information literature. It postulates that information about food availability or safety may not be readily obtained by “personal” inspection (Valone and Templeton 2002, Danchin et al. 2004, Dall et al. 2005). Under such circumstances animals may benefit from cues from conspecifics (Conlisk 1980, Clark and Mangel 1984, Stamps 1988, Valone 1993, Ruxton et al. 1995, Valone and Templeton 2002, Danchin et al. 2004, Dall et al. 2005, King and Cowlishaw 2007). Note that copying behaviour does not necessarily direct animals towards the most rewarding food patch (Beauchamp et al. 1997, Sirot 2006, Sumpter and Pratt 2009). Particularly, if predecessors selected a sub-optimal patch, copying may lead to a collective mistake (Beauchamp et al. 1997, Giraldeau et al. 2002). In the ecological literature it is widely acknowledged that a better understanding of the impacts of social benefits on foraging (and other types of) behaviour is needed (Fryxell 1991, Fletcher 2006, Nilsson et al. 2007, Jeanson and Deneubourg 2007, Campomizzi et al. 2008). However, empirical research on the impacts of resource availability, interference and conspecific attraction on the spatial distribution of foraging animals has been hampered by a noticable

15-03-2012

08:56

Pagina 53

lack of a coherent framework to measure conspecific attraction and interference (see also Nathan et al. (2008) who argue in a similar vein that lack of a framework hampers the understanding of movement). Here, we present an operational definition based on the notion that conspecific attraction will manifest itself as the tendency of conspecifics to locate in each others’ vicinity. (Note the similarity to (2008) who postulate that the structure of a movement path is a reflection of the basic processes that produced it.) This implies that it will show up as positive spatial dependence, i.e. the number of animals at one location is positively correlated with the number of animals at neighbouring locations (controlling for the effects of other location factors). Interference, on the other hand, has a depressing effect on positive spatial dependence, as it induces animals to keep some minimal distance from each other. Hence, both conspecific attraction and interference show up as spatial dependence, but in opposite directions. Interference drives animals away from each other and operates over short distances (Stillman et al. 2002, Vahl et al. 2005). Particularly, at small spatial scales, the effects of repulsion, due to attempts to reduce costs of interference, will show up as species specific minimal distances between individuals. In contrast, conspecific attraction operates over long distances and makes animals cluster. It follows that at large spatial scales (e.g. whole resource landscapes) the impact of conspecific attraction on spatial dependence will be more pronounced than the interference effect. Effects of exogenous environmental factors and spatial dependence in the response variable (i.e. spatial autocorrelation) can be estimated by means of spatial autoregressive (SAR) models which combine conventional regression models with a spatial autoregressive structure (Anselin 1988, LeSage and Pace 2009). Thus, estimates of the impacts of interference and conspecific attraction on the one hand, and of exogenous environmental factors on the other, may be obtained by means of estimating a SAR model on the basis of the spatial distributions of animals and food, respectively. The classical model is based on the assumption of an equilibrium distribution in which no animal has an incentive to find a better foraging site. However, in nature equilibrium distributions hardly, if ever, occur. In a group of foraging animals there are always animals in motion responding to, or anticipating, changing conditions. Particularly, incoming and relocating animals change both conspecific attraction and interference, and hence consumption rates, which may induce on-site individuals to relocate, which induces further relocation, and so on. In this paper we do not only consider the adequacy of measuring conspecific attraction by means of spatial autoregression in equilibrium distributions, but also in non-equilibrium distributions, denoted as “flocks in motion”. The specific objectives of this paper are the following. First, we develop an integrated model based on expected resource availability, interference and

ESTIMATION OF SELF-ORGANIZATION BY SPATIAL AUTOREGRESSION


53


15-03-2012

08:56

Pagina 54

conspecific attraction. Secondly, we simulate spatial distributions on the basis of the integrated model to illustrate how flocking patterns depend on, amongst others, interference and conspecific attraction and show that conspecific attraction decreases the predictability of patch selection. Thirdly, on the basis of simulated data, we analyse the applicability of SAR models to estimate the impact of resource availability, and the joint impact of conspecific attraction and interference, for equilibrium and non-equilibrium distributions. Fourthly, we show that the total (i.e. direct + indirect effects) impact of resource distribution on the spatial distribution of foragers can be estimated by the spatial multiplier. Fifthly, we show how the autoregressive approach can be applied in empirical research. Finally, we discuss some of the simplifying assumptions on which our model is based.

The model

CHAPTER 3

Our integrated classical-conspecific attraction model (integrated model for short) is based on the notion that the selection of a foraging site involves balancing costs and benefits. We assume that if an animal located too close to conspecifics, the costs due to the presence of conspecifics (i.e. interference) would exceed the benefits. The cost of interference and its repulsive effect, however, rapidly level off to zero when inter-individual distance increases (Stillman et al. 2002, Seppänen et al. 2007). This assumption is based on the notion that interference arises from behavioural interactions (e.g. stealing prey items or fighting) that are only possible amongst nearby animals (Vahl et al. 2005). Benefits due to the presence of conspecifics (relating to food availability and safety) decrease at a lower rate with distance than interference costs because individuals can generally observe and benefit from conspecifics that are relatively far away (Fernandez-Juricic et al. 2004). Hence, we assume that conspecific attraction operates over larger distances than repulsion. The above assumptions with respect to interference and conspecific attraction are corroborated by the spacing behaviour of e.g. sycamore aphids (Drepanosiphum platanoides) (Kennedy and Crawley 1967) and various species of shorebirds (Moody et al. 1997, Folmer et al. 2010).

54

Expected consumption rate We assume that the expected consumption rate at a foraging site (hereafter labeled “cell”) is a function of expected resource availability and interference. With respect to expected resource availability, we assume either perfect or imperfect information. Information uncertainty is incorporated by adding stochasticity to Beddington’s functional response model (Beddington 1975).


15-03-2012

08:56

Pagina 55

Denoting expectation of a stochastic variable by ´, the expected consumption rate, Cí for each cell i is: Cí =

aRí 1+ ahRí + qPi

where Rí is the amount of food an animal expects to find in cell i, Pi the number of conspecifics in cell i, q the interference parameter, a the attack rate and h handling time. In the simulations, we model information uncertainty for cell i by drawing from a left-truncated (at 0) normal distribution with mean the true amount of food and standard deviation θ. To simplify the simulations, but without loss of generality, we assume that a and h equal 1. Conspecific attractiveness and spatial dependence As noted above, the presence of conspecifics in a cell is taken by searching animals as an indication of food availability or safety. Hence, the presence of conspecifics signals the attractiveness of a location. Conspecific attractiveness of a given cell is a function of the number of its “own” conspecifics, i.e. conspecifics located within the borders of the own cell, and of the number of conspecifics in neighbouring cells. We assume that the conspecific attractiveness of cell i (Si) increases with the number of conspecifics up to an asymptote as follows:

Si = s

N

1+ ∑ Wij Pj

,

j

where N is the total number of cells in the spatial system and W is the spatial weights matrix representing the structure of the spatial system. We assume firstorder queen contiguity between cells. That is, Wij = 1 if cell i and j have a common border or vertex. Moreover, to allow for conspecific attraction within a cell, we define Wii = 1. Finally, Wij = 0 elsewhere. W is row-normalized, i.e. each element is divided by the sum of its row elements such that the sum of each row equals 1. By row-normalization, Si is independent of the number of neighbours N of cell i. The term ∑ Wij Pj indicates that animals are attracted to cell i because j of the presence of conspecifics in cell i and in the neighbouring cells. The intensity of conspecific attraction is given by the parameter s. Note that there are manifold reasons for animals to form groups. Rather than defining a specific attractiveness function, we simply assume that the nearby presence of conspecifics has functional advantages and that the attractiveness of a location relates to the numbers of conspecifics in its near surrounding.


N

∑j Wij Pj

55


15-03-2012

08:56

Pagina 56

Total attractiveness and leader-follower heterogeneity Total expected attractiveness of cell i, Tí , is the sum of its expected consumption rate (Cí ) and its conspecific attractiveness Si. N

Tí =

aRí 1+ ahRí + qPi

+s

∑j Wij Pj N

1+ ∑ Wij Pj j

In nature, animal populations are heterogeneous with respect to their ability to avoid predation and the knowledge they have of the distribution of resources (Stamps 1988, Krause and Ruxton 2002, van Gils et al. 2006). Hence, there is variation in benefits that individuals obtain from locating near conspecifics. This type of heterogeneity is modeled by having a fraction of the population that is insensitive to the “attraction signals” from their conspecifics (i.e. s = 0), viz. their choices are based on the expected consumption rate (Cí ) only. (In a producer-scrounger model the insensitive type would be considered a producer and the sensitive type an opportunistic scrounger.) Rands and Johnstone (2003) label the insensitive types “leaders”, and the sensitive types (s > 0) “followers”. We will use this terminology below.

CHAPTER 3

Setup of the simulations

56

In this section, we first describe how the different resource landscapes in which the foragers select patches, are generated. Next, we present the agent-based patch choice procedure including the properties of the foragers and (re)location rules based on the attractiveness of the cells. We consider a continuous resource landscape made up of 24 × 24 cells. The amount of food in each cell is generated by drawing a value from the standard normal distribution. Since this may lead to unrealistically large differences in values between adjacent cells, we smooth the landscape by means of a Gaussian 2D convolution filter (Oksanen and Sarjakoski 2005). This smoother preserves the original normal distribution. The level of smoothness is determined by the range of the kernel, r. In the simulations r takes the values 0 (no smoothing), 3 (intermediate) and 5 (strong smoothing). The final resource landscape is obtained by means of inverse standardization, i.e. each smoothed cell value is multiplied by the standard deviation 2 followed by adding the mean 5. Negative values (which because of the selected values of the moments of the distribution will be very rare) will be set at 0. The numbers of animals that forage in the resource landscape is varied such that the mean densities correspond to 1, 2 and 5 animals per cell. The parameter

15-03-2012

08:56

Pagina 57

s varies between 0 and 1 with steps of 0.1; q varies between 0.1 and 1 with steps of 0.1. With these parameter combinations the resulting distributions range from dense to sparse flocks (Fig. 3.1). The fraction of leaders is 0.1, and, hence, the rest of the population is made up of followers. Of course, when s = 0, there are no followers and all individuals find food individually and thus are “leaders”. Individual animals enter the landscape sequentially. Leaders and followers enter in random order. A leader selects a cell on the basis of the highest expected consumption rate Cí ; a follower on the basis of highest expected total attractiveness Tí . Information uncertainty on the food distribution (θ ) for a screening animal is 0 (perfect information) or 2 (incomplete information). (If the animals have perfect information there is no need to locate near conspecifics to reduce search costs. Hence, in the case of perfect information, reduction of costs is related to vigilance and risk dilution only.) We assume no food depletion so that food availability is the same for each individual. The simulations are agent-based; that is, for each individual we record its position in the landscape and whether it is a leader or follower. Moreover, the expected resource availability of each cell is registered. The number of replications per parameter combination (s, q, r, θ and animal density) is 25. For each replication a new food distribution is generated. Due to the arrival of new animals, which leads to increased interference and conspecific attraction, the expected and realized total attractiveness for animals already present change. Animals are able to respond to the changing conditions by relocating to other cells. Relocation is simulated as follows. When all the animals have entered the resource landscape, first the leader that is worst off in terms of realized consumption rate (Ci) relocates to a new cell, if it perceives an opportunity for improvement. The new location, j ≠ i, is chosen on the basis of the highest expected consumption rate (C´j ). Next the follower that is worst off in terms of realized total attractiveness (Ti), that sees opportunity for improvement moves to the cell where expected total attractiveness, Tí , is highest. Note that if relocation does not increase expected consumption rate or total attractiveness, the animals stay in their current cells. The relocation of a leader or a follower is labelled a relocation event. After the first relocation event, the other animals have the opportunity to relocate. The relocation process continues until equilibrium is obtained, i.e. all leaders and followers are in the most attractive cells and have no incentive to move. Individuals remember the resource availability of the visited cells, i.e. Rí is updated after each relocation by removing the stochasticicty (associated with imperfect knowledge) of the visited cells. (Obviously, the memory assumption is only relevant for the imperfect information case.) The number of relocation events to obtain equilibrium will be recorded. If no equilibrium is reached within 2000 relocation events, the run is terminated. Note that because of the memory assumption, equilibrium will ultimately emerge.



57


15-03-2012

08:56

Pagina 58

Evaluation methods Spatial autoregression The spatial autoregression (SAR) model combines a conventional regression model with a spatial autoregressive structure. The SAR model estimates the impacts on the number of animals in a cell of the exogenous environmental variables and of the number of animals in neighboring locations weighted by the spatial weights matrix. That is, the SAR model for our integrated model, for cell i = 1, 2,…, N, reads: yi = β 0 + β 1 xi + ρ∑ j Wij yj + ε i

CHAPTER 3

with yi = ln(number of animals +1), xi = ln(amount of food +1), W is defined as above, except that Wii = 0 (see below). Row-normalization implies that for N cell i, ∑ Wij ln(Pj +1) is the average (transformed) number of conspecifics in its j first order contiguous cells. The regression coefficient β 0 is the intercept, β 1 represents the direct food effect and ρ is the spatial autoregressive coefficient. The number of animals and the amount of food are increased by 1 because of possible zero arguments of the logarithm functions. The spatial autoregressive model will be estimated by means of maximum likelihood (ML). See Anselin (1988) and LeSage and Pace (2009) for details. In contrast to data generation where Wii = 1, Wii = 0 in the SAR model. This is because for a single cell there is no spatial dependence by definition (i.e. spatial dependence is between cells only). This does not mean that there could be no conspecific attraction within a given cell. However, to estimate within-cell attraction with a SAR model, spatial disaggregation (splitting up a cell in smaller parts) is required.

58

Evaluation of the adequacy of the SAR framework for analysis of equilibrium distributions and flocks in motion We evaluate the adequacy of the SAR framework to estimate the effect of food availability and the joint effect of conspecific attraction and interference for equilibrium distributions and flocks in motion as follows. First, since perfect information always leads to equilibrium, we only consider the case of information uncertainty. Secondly, for a given parameter configuration, we only consider replications for which in one subset of the 25 replications equilibrium distributions were achieved and for the complimentary set flocks in motion. The performance of the SAR framework is evaluated by comparing the differences between the average spatial autoregressive coefficient of equilibrium distributions (ρeq) and the average spatial autoregressive coefficient for flocks in motion (ρmotion) in the two subsets. Thirdly, the number of relocation events


15-03-2012

08:56

Pagina 59

Spatial multiplier Not only do animals in a given cell i attract animals to surrounding cells but also vice versa: animals in surrounding cells attract animals to cell i. Specifically, the first individual that settles in the landscape selects a cell i on the basis of its expected consumption rate (direct food effect). Its presence at cell i induces other animals to select cell i or neighbouring cells (first order indirect food effect) which induces other animals to select cell i or neighbouring cells (second order indirect food effect), and so on. Hence, food availablity in cell i does not only impact on the number of animals in cell i, but also in other cells. In other words, food availability in cell i “multiplies” through the spatial system. The average total food effect (direct + all indirect effects) on the number of animals is obtained by multiplying the food regression coefficient β1 in the SAR model, denoted β1SAR, by the spatial multiplier 1/(1- ρ). In other words, the spatial multiplier is the average level by which the direct effect of a factor is multiplied to account for indirect effects in the system to obtain the total effect (LeSage and Pace 2009). Note that estimating the effect of food availability on the number of foraging animals per cell by ordinary least squares (OLS) (which ignores spatial spillover, i.e. the explanatory variable ∑ j Wij yj is omitted), is biased because the effect of conspecific attraction is ignored (Beale et al. 2010). Specifically, β1OLS, the regression coefficient of food availability estimated by OLS, is biased, i.e. it over-estimates the direct food effect.

Simulation results Integrated equilibrium distributions To get insight into the impacts of various components of the integrated model, especially conspecific attraction and interference, we present various equilibrium distributions in Figure 3.1. This figure shows that for a fixed resource landscape the spatial distribution of omniscient animals (θ = 0) with 10% leaders varies with conspecific attraction and interference. (Observe that conspecific


until equilibrium emerges varies. To get insight into the speed of emergence of an equilibrium (if it emerges) as well as the stability of ρmotion over relocations, we report by classes of relocation events the number of equilibria (u) that have emerged, and the number of flocks in motion (v) as well as the values of ρeq and ρmotion. The classes are: 1 – 500, 501 – 1000, 1001 – 1500 and 1501 – 2000. We use classes because the number of relocation events for which an equilibrium emerges varies for a given parameter combination. Finally, several parameter configurations will be considered.

59

s = 0.5

s=0


15-03-2012

08:56

Pagina 60

q = 0.1

q = 0.5

q=1

● ●●●●●●●● ●●●● ●●●●●●●●●●● ●●●●● ●●●●●● ●● ●●●●● ●● ● ●●●●●● ● ●●●●● ●●● ● ●● ●●● ●● ●● ● ●●●●●●●●● ●● ●●●●●●●●● ●● ● ●●●●●●●● ●● ● ●●●●● ● ●●●● ● ●●●● ● ●● ● ●●● ●● ● ●●● ● ●●●● ●●●●●●●● ●●●●● ●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●● ●●●●●● ●● ●●●●● ●●●●●● ●●●●● ●●●● ●● ●

●●● ●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●● ●●●● ●●●●●●●● ●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●● ●●●● ●●●● ●●● ● ●●●●●● ●●●● ● ●● ●●●●● ●●●● ●●● ● ●●●● ●●●●● ●●●● ●●●● ●●●●●●●●●●● ●●●●●● ● ●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●● ● ●●●●●●●●●●●●● ●●●●●●● ●●●●●●●● ●●●●● ●●●●●● ● ●●●●●●●●● ●●●● ●●●●●● ●●●●● ●●●●●●●●●●●●● ●●●●● ●● ●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●● ●●●●●● ●●●● ●●●●●●●●● ●●●●●● ●●●●●●●● ●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●● ●●●●●●● ●●●●●●●● ●●●●● ●●●●●● ●●●●●● ●●●● ●●●●●●● ●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●● ●●●●●● ● ●●●●●●●●●●●●● ●●●●●● ● ●●●●●●●●●● ●●●● ●●●●●● ● ●●●●●●● ●●●●●●●●●● ●●●● ●●●●●● ●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●

●●●●●●●● ●●●●●●●●● ●●●● ● ●●● ●●

● ● ●●●●●●●●● ● ●● ●●●●● ●●●●●●●● ●●●●● ●●●●●●●●●●● ●●●●● ●●● ●●●●●● ●● ●●●●●● ●●●●●●●● ●● ●●●●● ●●● ●●● ●●●●●●● ●●● ●●● ●●●●●●●●●● ●●● ●●●●●●●●●●● ●●● ●●●●●●●●●● ●●● ● ●●●●● ●● ● ●●●●● ●● ●●●● ●●● ●●●● ●●● ●●● ● ●●●●●●●●● ●●●● ●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●● ●●●●●● ●●● ● ●●●●● ●●●●● ●●●● ● ●●●● ● ●●

●●● ●●●

●● ●●●● ●●● ●

●● ●●●● ●●●●●

●●●●●● ●●●●●● ●●●●●● ● ● ●● ●● ●

●● ●● ●● ● ●● ●

● ●● ●●

● ●● ●

●

●●●● ●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●● ●●●●

●●● ● ●●

●● ●● ●● ●

●●● ●●●●● ●●●●●● ●●●●●● ●●●●●● ●●●●

● ●●

●●● ●●●● ●●●● ●●

●● ●● ●● ● ●

s=1

●● ●● ●●● ●●●● ●●●●●●●●● ●●●●●●●● ●●●●● ●●●● ● ●●● ●● ● ●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●● ●●●●

●

●● ●●

●●●

●● ●●

● ●

●● ●● ●

●●

●●●●●●●●

●●●●●●●

●●

●●● ● ●●●●● ●●●●

●● ●●● ●●●●●● ●●●●●

●●●● ●●●●

●●● ●●●● ●●●● ●●

●● ●●●● ●●●● ●●● ●● ●● ●

●●●●●●●● ●●●●●●●●●● ● ●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●● ●●●●● ●●●●

●●

●

●● ●● ●●●●●● ●●●●●

● ●

●●●● ●●●●

●● ●●●● ●●● ●●● ●● ●●

food 0 2 4 6 8 10

animals ●

5

●

10

●

15

●

20

● 25

● 30 ● 35

● ●●●● ●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●

●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●

CHAPTER 3

Figure 3.1. Spatial distribution of omniscient animals (θ = 0) in a fixed and smoothed (r = 5) 24 × 24 cells resource landscape with mean animal density equal to 1 individual per cell. Each panel is based on one simulation run for a given combination of s (conspecific attraction parameter) and q (interference parameter). Colour intensity of the landscape represents food density and the size of the dots represent the number of animals.

60

attractiveness in this case relates to decreased vigilance costs and dilution of predation risk only.) Conspecific attraction results in clustering of animals, whereas interference drives individuals away from each other to cells with lower food availability. Strong interference leads to a spatial distribution of foraging animals that reflects the food distribution more closely. Strong conspecific attraction in combination with weak interference results in dense clusters, whereas strong interference combined with weak conspecific attraction leads to sparsely populated cells and spaced-out distributions.

15-03-2012

08:56

1

2 ● ●

●

Pagina 61

● ●● ● ● ● ●● ● ● ●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●● ● ● ●●● ●

●

3 ●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

● ● ● ● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●●● ● ●

● ●●● ● ●●● ●

● ●

● ● ● ●

● ●

●

● ●●● ● ● ● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

● ● ●

●

●

●

●

●

●

● ●

●

●

● ● ● ● ● ●

● ●

● ● ●

● ●●● ● ●●● ● ●●● ●

●

●

●

●

food

● ● ●

0 2

●

●

●

● ●

● ●●● ● ● ●● ● ●

●

●

●

●

●

●

●

●

●

●

4

●

●

6

● ● ●

8

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●● ●●● ● ●

● ● ● ● ● ●● ●

● ● ●

●

●

●

●

●

●

●

●

● ● ● ● ●

● ● ● ● ●

●

●●●

● ●● ● ● ●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ● ● ● ●

● ●

●

● ● ●●● ● ●

●

●

●

●

●

●

● ●

● ●● ● ● ● ●● ● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●

● ● ● ● ● ●

●●● ● ● ● ● ●●● ● ● ● ● ●

● ● ● ●

●

●

●

●

● ●●●

● ●● ● ● ●●● ●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

● ● ● ● ● ●

●●● ●●● ●

●

animals 5 ●

●

10

●

15

●

20

●

25

●

●

●

●

● ● ●

●

●

● ● ●

9 ●

●

● ●●● ● ●● ● ●

●

●

●

●

●

●

●

● ●

●

●

● ●

●

●

●

●

●

●

10

●

●

8 ●

●

●

●

6

●

●

●

●

●

●

●

●

●

● ●

●

●

7 ●

●

●●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●● ●

●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

● ● ●

● ● ●● ● ● ●● ● ●

●

●

●

● ●●● ● ● ●● ● ●

●

●

●

●

5 ●

●

● ● ●● ● ● ●● ●

●

●

●

●

4 ●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

● ●

●●● ●●●

●

●

●

●

●

●

●

● ● ● ● ● ●● ● ●

● ● ● ● ● ●

●●●● ● ●● ● ● ●●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

● ●

● ● ● ● ●

● ● ●●● ● ●

● ●●● ● ●● ● ●

● ● ●

Figure 3.2. Equilibrium distributions of animals that are interference-insensitive (q = 0.1), moderately sensitive to conspecific attraction (s = 0.5) and with information uncertainty on the food distribution (θ = 2). Each panel shows the outcome of one replication. The 24 × 24 cells resource landscape is fixed over replications; r = 5; animal density = 1 cell-1. See Figure A1 for further details.

Figure 3.2 presents the distributions of 9 independent replications for a fixed resource landscape with animals that are moderately sensitive to conspecific attraction (s = 0.5) and interference insensitive (q = 0.1). The replications only differ by information uncertainty about the food distribution (θ = 2). The figure shows that the location of the clusters of animals is highly variable and unpredictable under these circumstances. Note that the distributions presented in Fig. 3.2 apply to equilibrium distributions obtained after varying numbers of relocations. When the number of relocations increases, the resulting equilibrium



61


15-03-2012

08:56

Pagina 62

distributions become more similar to each other because of memory, i.e. individuals remember the resource availability of the visited cells. Note, however, that although they become more similar, they still tend to differ from each other (see Appendix A for further details). The spatial autoregressive coefficient for equilibrium situations (ρeq) and flocks in motion (ρmotion) Before going into detail, we make the following observations. First, there are quite a large number of zero counts in the data. This type of data can be analysed by zero-inflated SAR models (Agarwal et al. 2002). However, we Table 3.1. Average ρeq and ρmotion for animal density 1 and r = 0 for selected values of q and s. relocations q s 0.2

0.4

0.6

0.4

0.4

0.6

0.6

0.0

0.2

0.8

0.0

0.2

CHAPTER 3

1.0

62

0.0

1 – 500 ρmotion ρeq

501 – 1000 ρeq ρmotion


0.86 (0.00) v=20 0.88 (0.00) v=16

0.85 (0.00) v=20 0.87 (0.00) v=16

0.86 (0.02) u=3

0.84 (0.00) v=20 0.87 (0.00) v=16

0.82 (0.00) u=5 0.87 (0.01) u=6

0.84 (0.00) v=20 0.87 (0.00) v=16

0.85 (-) u=1 0.89 (0.01) u=5

0.85 (0.00) v=18 0.89 (0.01) v=5

0.86 (0.01) u=2 0.89 (0.00) u=7

0.85 (0.00) v=18 0.89 (0.00) v=5

0.84 (0.00) u=4 0.89 (0.00) u=8

0.85 (0.00) v=18 0.89 (0.01) v=5

0.56 (-) u=1

0.00 (0.01) v=24 0.58 (0.01) v=14

0.58 (0.04) u=2

0.01 (0.01) v=24 0.54 (0.01) v=14

-0.09 (-) u=1 0.56 (0.01) u=8

0.01 (0.01) v=24 0.52 (0.01) v=14

-0.01 (0.01) v=14 0.48 (0.03) v=4

0.45 (0.01) u=5

-0.01 (0.01) v=14 0.42 (0.03) v=4

0.00 (0.03) u=4 0.46 (0.01) u=9

-0.02 (0.01) v=14 0.42 (0.02) v=4

0.00 (0.01) u=7 0.45 (0.02) u=7

-0.02 (0.01) v=14 0.42 (0.02) v=4

-0.02 (0.02) v=8

0.03 (-) u=1

-0.01 (0.01) v=8

0.02 (0.02) u=9

0.02 (0.01) v=8

-0.02 (0.01) u=7

0.01 (0.01) v=8

0.86 (0.00) v=18 0.90 (0.01) v=5 0.00 (0.01) v=24 0.65 (0.01) v=14


Notes- u and v: the number of equilibrium distributions and flocks in motion in an interval of relocations, respectively. Standard errors are within parantheses. Averages and standard errors rounded off. (-): no standard error because of one replication only.

15-03-2012

08:56

Pagina 63

found the residuals to approach normality (although they are a bit peaked). Given these characteristics and the purpose of the paper to demonstrate SAR modelling in general, we apply the standard SAR model. However, in empirical applications zero-inflated SAR models may be considered in the case of many zero counts. Secondly, as explained above, comparison of ρeq and ρmotion is only relevant for the imperfect information case because under perfect information there will always be an equilibrium. Thirdly, consider Table 3.1. To economize on space we do not report results for s >0.6 as they are virtually identical to those for s = 0.6. For the same reason we only consider increases of s and q by steps of 0.2, animal density 1 and r = 0. The results for other values of these parameters are very similar to the results presented in Table 3.1. A more complete table is presented in Appendix B. From Table 3.1 the following conclusions emerge: (1) ρeq and ρmotion hardly differ in most cases. In approximately 85% of the cases the difference is within a two-sided 95 % confidence interval, in approximately 95% of the cases in a two-sided 90% confidence interval; (2) the number of relocations hardly affects the development of the spatial autoregressive coefficients, except in the case of ρmotion for q >0.6. After relatively large changes for less than 1000 relocations, this coefficient stabilizes; and (3) large values of q have a depressing effect on the spatial autoregressive coefficients. This finding follows from the fact that interference induces animals to stay away from each other. The above implies that the spatial autoregressive coefficient reflects the impacts of conspecific attraction and interference nearly equally well for equilibria and for flocks in motion. This finding is further supported by the stability of the spatial autoregressive coefficient over relocations. Given the above findings there is no need to distinguish between ρeq and ρmotion. We generically refer to them by ρ. Hence, the analyses below are based on the combined subsets, i.e. the entire set of ρ values. The spatial autoregressive coefficient as function of the simulation parameters In this subsection we pay attention to the spatial autocorrelation coefficient as a function of r, animal density and, especially, s and q. As observed in the previous subsection, the results in Table 3.1 hint at a logistic relationship between ρ and its determinants, especially s. Inspired by the results in Fig. 3.1 and Table 3.1, we also included the interaction term s × q in the model. Therefore, we estimate the following model: 1n(

ρ ) = β 0 + β 1 s + β 2 q + β 3 density + β 4 r + β 5 sq + ε 1– ρ

with ε the disturbance term and the other variables as defined above.



63


15-03-2012

08:56

Pagina 64

Separate regressions are performed for perfect and imperfect information. Before going into detail, we observe that in ~2% (1094 out of 49500) of the cases a negative ρ with mean –0.037 was obtained. Because of the logit transformation of ρ, these cases were removed from the dataset. Table 3.2 shows a high overall goodness of fit for each of the two models, as expressed by the R2. It also shows that the signs of the coefficients of the explanatory variables of the two models are the same and in line with expectations: ρ increases with s and declines with q while animal density has a negative impact, since crowding drives animals to other cells. The coefficient of the interaction term s × q is positive because s induces animals to locate in each others’ vicinity while q restricts the number of animals within cells which leads to large, but sparse flocks. Sparse flocking renders neighbouring cells more similar and thus leads to stronger spatial dependence. An interesting finding is that under imperfect information, the coefficient of s is larger, and the coefficient of q in absolute value is smaller than in the case of perfect information. From the above it follows that ρ is a function of s, q, r and animal density and varies by information uncertainty. Hence, for given values of q, r and animal density, ρ is a function of s only. Similarly, ρ is a function of q only, if all other variables are known. An important issue in empirical applications is the sensitivity of the estimates to the scale of observation (Wiens 1989, Levin 1992). As pointed out above, conspecific attraction and interference may occur at various scales. To capture their impacts by means of spatial dependence, the scale at which data on the distribution of animals are available should approximately match the behavioural scales. These scales, however, may not be evident beforehand. We tested how our findings depend on the scale of observation by aggregating adjacent patches and performing the same analysis. The results obtained show that the coefficient of s increases substantially when going from level (1 × 1) to level Table 3.2. 1n(

ρ ) as a function of the simulation parameters. 1– ρ perfect information

imperfect information

Constant

1.67 (0.02)

1.49 (0.02)

s

2.39 (0.03)

2.53 (0.03)

q

–3.12 (0.03)

–2.40 (0.03)

sxq

CHAPTER 3

Animal density

64

r R2

2.53 (0.05)

1.74 (0.05)

–0.63 (0.00)

–0.56 (0.00)

0.29 (0.00)

0.27 (0.00)

0.86

0.82

Note- predictors: conspecific attraction (s), interference (q), animal density resource and smoothness (r). Coefficients and standard errors (within parentheses) are rounded off.


15-03-2012

animal density : 1

08:56

Pagina 65

animal density : 2

animal density : 5

8

OLS SAR SPM

coefficient

6

4

2

0 0.0

0.5

S

1.0

0.0

0.5

S

1.0

0.0

0.5

1.0

S

(2 × 2) while it is constant (perfect information) or decreases slightly (imperfect information) from (2 × 2) to (4 × 4). For the coefficient of q there is a monotonic decrease (in absolute value) for both information cases. For animal density and r there are only minor changes, although for the latter there is a change of sign when aggregating (Appendix C). The decline of the coefficient of q for increasing scale follows from the fact that the impact of interference on the spatial distribution of foragers declines when distance increases, as pointed out in the Introduction. The increase of the coefficient of s follows from the fact that when going from level (1 × 1) to level (2 × 2) the similarity, between contiguous regions in terms of foraging animals, increases. However, when going from level (2 × 2) to (4 × 4) the similarity decreases. This development of the coefficient of s reflects the tendency for conspecific attraction to level off when distance increases beyond a threshold. The upshot is that the autoregressive model adequately reflects conspecific attraction and interference at various levels of aggregation and makes it possible to gain insight into their separate effects. The spatial multiplier In this subsection we address the consequences of ignoring conspecific attraction on the estimator of the food effect. We first compare β1SAR obtained from the SAR model containing both the exogenous variable xi = ln(amount of food


Figure 3.3. Estimated direct and total food effects for an intermediate level of interference (q = 0.5), intermediate smoothness (r = 3), conspecific attraction (s) as indicated in each panel, animal densities 1, 2, 5 and θ = 0. The 95% confidence intervals are indicated by bands. Confidence intervals for β1SAR and β1OLS are very narrow and therefore do not come through everywhere. OLS = Ordinary Least Squares model; SAR = Spatial Autoregressive model; SPM = Total average food effect based on the SAR regression coefficients and the spatial multiplier. Similar results hold for other values of q and r.

65


15-03-2012

08:56

Pagina 66

+1) and the autoregressive term, with β1OLS obtained from the model with xi only and estimated by ordinary least squares (OLS). Figure 3.3 shows that for s fixed at zero, the OLS and SAR coefficient estimates are equal. However, when s is greater than zero, β1OLS exceeds β1SAR for all animal densities. Hence, ignoring conspecific attraction leads to overestimation of the direct effect of resource availability. Figure 3.3 also shows that β1SAR declines with increasing conspecific attraction for all animal densities which reflects that due to increasing conspecific attraction, the direct impact of food availability decreases. As observed above, β1SAR only gives the direct impact of food availability, not the total effect, i.e. direct effect plus indirect effects. To obtain the (average) total food effect β1SAR is multiplied by the spatial multiplier. Fig. 3.3 shows that the total effect predicted by the SAR model exceeds the total effect predicted by the OLS model by an order of magnitude. Hence, the interdependent nature of animals reinforces the impact of resource availability on their foraging distribution.

CHAPTER 3

Discussion

66

Foraging distributions of social animals are complex phenomena. One of the reasons for our poor understanding of the animal-habitat relationships is the unobservable nature of conspecific attraction and interference. In this paper we have shown that they may be measured indirectly by way of their manifestations, i.e. the tendency of conspecifics to locate in each others’ vicinity or to stay away from each other, respectively. Conspecific attraction shows up as positive spatial dependence, whereas interference has a depressing effect on it. We have shown that standard statistical methods that ignore spatial dependence (particularly OLS), will give biased estimators and tests of relationships between animals and their resources when they are influenced by conspecific attraction (also see Dormann et al. 2007; Beale et al. 2010). We have presented the spatial autoregressive model as an adequate alternative that gives unbiased results. Since the estimated spatial autoregressive coefficient is influenced by interference and conspecific attraction as well as animal density, information uncertainty and smoothness of the resource landscape, prior information on these variables is needed to derive estimates of self-organization, i.e. the combined effect of conspecific attraction or interference. In addition, if prior information on conspecific attraction (interference) is also available, the separate effect of interference (attraction) can be obtained. In empirical research, this kind of prior information is sometimes available or can be readily obtained. For example, in shorebird foraging environments information on resource availability and functional responses are available or can be relatively easily collected. For instance, red knots (Calidris canutus) are known to be rather interference-insen-

15-03-2012

08:56

Pagina 67

sitive (van Gils and Piersma 2004), and oystercatchers (Haematopus ostralegus) interference–sensitive (Vines 1980). Information on the other relevant parameters, i.e. animal density and resource density, can be obtained from spatial population counts and food surveys such as the long run monitoring programs for the Dutch Wadden Sea (van Gils et al. 2006; Kraan et al. 2009; Folmer et al. 2010). In species distribution datasets there often are many zeros for the dependent variable (Martin et al. 2005). This characteristic is often taken as an indicator of the fact that the species under consideration tends to locate in the most suitable habitat which itself is spatially heterogeneously distributed. Here we have proposed a supplementary explanation. Particularly, we have shown that conspecific attraction may lead to large areas with abundant food remaining unoccupied. We did not investigate the effects of varying attack rates and handling times on the spatial distribution of foragers. However, their effects in our model can be derived from van der Meer and Ens (1997) who show that in Beddington’s functional response model (1975) an increasing attack rate will lead to more foragers in rich food patches and that increasing handling times have an opposite effect. An increasing attack rate will have similar effects in our model resulting in smaller but denser flocks. Moreover, the attractiveness of surrounding cells will also increase. Thus, conspecific attraction will further amplify the direct effect of increasing attack rate. Increasing handling time on the other hand will decrease the density of foragers per cell leading to a larger fraction of sparsely populated cells. In this case conspecific attraction will be directed towards a larger number of cells which will lead to spaced-out flocks. This effect is similar to the interference effect. The simulation results confirm the hypothesis that the impacts of conspecific attraction and interference are virtually the same for equilibrium distributions and flocks in motion, and that their impacts on the distributions of flocks in motion are stable when animals relocate. The explanantion for this finding is that when selecting a foraging patch, an animal is driven by its knowledge of the resource landscape and by the signals from conspecifics. Whether searching is at an early stage or at the final stage is irrelevant in our model; it is always the same mechanisms that drive patch selection. The result implies that analyzing the impacts of interference and conspecific attraction does not require equilibrium distributions. This is an important result for empirical applications because in nature flocks virtually always are in motion. Another interesting finding is that under information uncertainty, the effect of conspecific attraction on the spatial distribution is larger than under perfect information, in spite of possible higher interference costs. The explanation is that compared to perfect information, the average expected benefit of the presence of conspecifics increases or the expected costs of interference decrease, such that the animals tend to cluster more and spatial dependence increases.



67

CHAPTER 3


68

15-03-2012

08:56

Pagina 68

The simulation results imply that if the average consumption rate is not substantially affected by the presence of conspecifics (low interference), populations may benefit from independent individuals (leaders), as long as they select optimal food cells. Resource-rich areas will then be increasingly occupied which decreases the mismatch between the resource and animal distributions. It has been shown that leadership status may be driven by energy stores (Rands et al. 2003), body size (Krause et al. 1998), age or dominance (Krause and Ruxton 2002), i.e. hungry, large, mature or dominant individuals taking leadership. This implies that heterogeneity amongst interdependent individuals may be beneficial to the population. The model and simulations presented in this paper focuses on conspecific attraction and interference and abstract from various other factors and processes that also affect the spatial distribution of foragers. For example, we assumed that the proportions of leaders and followers are fixed and that all followers are equally sensitive to conspecific attraction, i.e. all followers having the same s. A more realistic approach would be to have s follow a continuous distribution the shape of which affects the spatial distributions of the foragers. For instance, for the same range, a distribution of s with a fat right tail will result in a more clustered spatial distribution of foragers than when s is drawn from a uniform distribution. It should be noted that drawing s from a continuous distribution rather than from the simple distribution applied above would not basically change the simulation procedure. In natural populations there are various factors (e.g. energetic state, information, predation risk) that simultaneously affect an animal’s level of conspecific attraction which, moreover, may vary for different timescales. The frequency distribution of conspecific attraction of a natural population will, thus, more likely be dynamic rather than fixed. For instance, aggregations of foragers may attract predators which in their turn, ceteris paribus, may increase the level of conspecific attraction between the foragers on a short timescale. On longer timescales, increasing age and experience may decrease the level of conspecific attraction. An interesting extension to our model would be to allow for temporal heterogeneity in conspecific attraction of individuals. Yet another process that may affect foraging distributions (but was ignored here) is resource dynamics. Particularly, spatially heterogeneous depletion and growth of resources is likely to negatively affect the predictability of optimal foraging locations and thus the spatial distribution of foragers and delay or prevent the emergence of equilibria. Our results indicate that in the case of varying resource densities, ceteris paribus, the impact of conspecific attraction on the spatial distribution and the importance of leaders will increase. (It should be noted, however, that through time, animals may also learn the spatial distribution of resources which decreases the role of attraction and the importance of

15-03-2012

08:56

Pagina 69

leaders.) However, on short timescales (i.e. the timescales at which flocking takes place) depletion of resources will often be small and the impact on foraging distributions negligible. For instance, in productive intertidal mudflat systems such as the Wadden Sea, where shorebirds forage on benthic prey during low tide, resources are depleted at slow rates. Due to their high reproductive rates, the densities of benthic animals build up fast during spring and summer (Beukema et al. 2002, Philippart et al. 2010) whereas depletion occurs at a much slower rate during late summer, autumn and winter when shorebird predators are around in greatest numbers (van de Kam et al. 2004). For this reason changes in benthos densities have been difficult to detect over time spans shorter than a month (Zwarts et al. 1992, Piersma et al. 1993, Kraan et al. 2009). The resource landscape for foraging shorebirds can, at least in the present case, be considered constant over short timescales. For other systems, resource depletion may be relevant, even in the short run of one visit. Over longer time spans, however, resource depletion is an important issue. Further development of our understanding of the long run spatial distribution of social foragers in relation to their resources requires integration of social foraging behaviour and resource dynamics. In this context, the role of leadership, learning and interdependency of foragers become important research themes. We expect that the foraging model and the proposed statistical methodology presented in this paper may play a role in this context. In conclusion, we concur with Lima and Zollner (1996) and Nathan et al. (2008) that research on habitat selection at the landscape-scale will benefit from research on animal behaviour on the micro-scale. We also share their conclusion that the main obstacle to the development of population habitat selection models is limited by poor knowledge about the information that animals have about landscape properties. Currently, the theoretical and empirical literature in this area is growing rapidly (van Gils et al. 2006, Rogers et al. 2006, Wikelski et al. 2007). Our study contributes to this literature by showing how conspecific interaction impacts on habitat selection in that it presents an operational definition of the unobservable processes of conspecific attraction and interference and develops a methodology that can disentangle them. It also contributes to research that focuses on the causes and consequences of interdependency among animals in general which is currently an important theme in behavioural research (Couzin et al. 2005, Conradt et al. 2009, Ramseyer et al. 2009, Sumpter 2010). Acknowledgements We thank Emile Apol, Thor Veen, Jan A. van Gils, Rampal Etienne, Joost Tinbergen, Ben Bolker and two anonymous reviewers for valuable discussion, comments and advice.



69


15-03-2012

08:56

Pagina 70

Appendix A: Development of spatial distributions due to relocations under imperfect information Figure A1 presents the development of the spatial distributions due to relocation for equilibrium distributions 3, 4, 5 and 7 in figure 3.2. Observe that for replications 4 and 7 equilibria are obtained between 1000 and 1500 relocation which implies that the distributions do not change after 1500. In replications 3 and 6 equilibrium is obtained between 1500 and 2000 relocations.

4

5

7

0

3

500

food 0 2 4 6

1000

8 10

animals 5 10 20 25

CHAPTER 3

2000

1500

15

70

Figure A.1. Development of the spatial distributions 3, 4, 5, 7 presented in Fig 3.2 due to relocations. Animals are interference-insensitive (q = 0.1), moderately sensitive to conspecific attraction (s = 0.5) and have imperfect information on the food distribution (θ = 2). Each column represents the spatial distribution after 0, 500, 1000, 1500 and 2000 relocations. The 24 × 24 cells resource landscape is fixed for the 4 replications; r = 5; animal density = 1 cell-1. Equilibrium distributions are plotted in black; flocks in motion are grey.


15-03-2012

08:56

Pagina 71

Appendix B: Development of spatial autoregressive coefficients for equilibrium distributions and flocks in motion

Table B.1. Average ρeq and ρmotion for animal density 1 and r = 0. relocations q s 0.2

0.0

0.2

0.4

0.6

0.8

1.0





0.00 (0.01) v=25 0.80 (0.00) v=25 0.86 (0.00) v=20 0.88 (0.00) v=16 0.89 (0.00) v=6 0.89 (-) v=1

-0.01 (0.01) v=25 0.78 (0.00) v=25 0.85 (0.00) v=20 0.87 (0.00) v=16 0.89 (0.00) v=6 0.88 (-) v=1

-0.01 (0.01) v=25 0.77 (0.00) v=25 0.84 (0.00) v=20 0.87 (0.00) v=16 0.88 (0.01) v=6 0.88 (-) v=1

-0.01 (0.01) v=25 0.75 (0.00) v=25 0.84 (0.00) v=20 0.87 (0.00) v=16 0.88 (0.01) v=6 0.88 (-) v=1

0.89 (0.01) u=5 0.89 (0.00) u=8

0.86 (0.02) u=3 0.88 (0.01) u=6 0.89 (0.00) u=11

0.82 (0.00) u=5 0.87 (0.01) u=6 0.88 (0.01) u=8 0.89 (0.01) u=5


This appendix presents a more complete version of Table 3.1 in the main text. Table B1 shows the average spatial autoregressive coefficient for equilibrium distributions (ρeq) and the average spatial autoregressive coefficient for flocks in motion (ρmotion). A “flock in motion” is a distribution for which no equilibrium was obtained within 2000 relocations. (Due to the memory assumption all distributions will eventually become equilibrium distributions.) Averages are based on the number of replications for which equilibrium and flocks in motion are obtained, respectively. To get insight into the evolution of ρeq and ρmotion these statistics are reported by classes of relocations: 1 – 500, 501 – 1000, 1001 – 1500 and 1501 – 2000. Classes are used because the number of relocations for which equilibrium emerges varies. Comparison between autoregressive coefficients is not always possible because for some parameter combinations either no equilibrium was reached for any of the 25 replications in which case all distributions are flocks in motion or all 25 replications reached equilibrium. The basic conclusions that emerge from table S2 is that the spatial autoregressive coefficients are virtually the same for equilibrium distributions and for flocks in motion. Additionally, ρmotion decreases slightly due to relocations. The decrease of ρmotion is strongest after the first relocations.

71


15-03-2012

08:56

Pagina 72

Table B.1. Continued relocations q s 0.4





-0.03 (0.01) v=25 0.76 (0.01) v=25 0.86 (0.00) v=18 0.90 (0.01) v=5 0.90 (0.01) v=4

-0.03 (0.01) v=25 0.71 (0.01) v=25 0.85 (0.00) v=18 0.89 (0.01) v=5 0.90 (0.01) v=4

-0.03 (0.01) v=25 0.68 (0.01) v=25 0.85 (0.00) v=18 0.89 (0.00) v=5 0.90 (0.01) v=4

-0.03 (0.01) v=25 0.65 (0.01) v=25 0.85 (0.00) v=18 0.89 (0.01) v=5 0.90 (0.00) v=4

0.0

0.2

0.4

0.6

0.8

0.90 (0.01) u=2

1.0

0.6

0.0

0.00 (0.01) v=24 0.65 (0.01) v=14

0.2

0.4

0.6

0.8

CHAPTER 3

1.0

72

0.85 (0.00) u=2 0.90 (0.01) u=4 0.91 (0.00) u=8 0.91 (0.00) u=11

0.85 (-) u=1 0.89 (0.01) u=5 0.90 (0.00) u=11 0.91 (0.00) u=11

0.56 (-) u=1 0.86 (0.00) u=10 0.90 (0.00) u=17 0.90 (0.00) u=15 0.91 (0.00) u=13

0.00 (0.01) v=24 0.58 (0.01) v=14

0.86 (0.01) u=2 0.89 (0.00) u=7 0.90 (0.00) u=7 0.91 (0.00) u=9

0.58 (0.04) u=2 0.84 (0.00) u=10 0.89 (0.01) u=3 0.90 (0.01) u=2 0.91 (-) u=1

0.01 (0.01) v=24 0.54 (0.01) v=14

0.84 (0.00) u=4 0.89 (0.00) u=8 0.89 (-) u=1 0.90 (0.00) u=5 -0.09 (-) u=1 0.56 (0.01) u=8 0.84 (0.00) u=3) 0.89 (-) u=1

0.91 (-) u=1

0.01 (0.01) v=24 0.52 (0.01) v=14


15-03-2012

08:56

Pagina 73

Table B.1. Continued

0.8



-0.01 (0.01) v=14 0.48 (0.03) v=4

-0.01 (0.01) v=14 0.42 (0.03) v=4

0.00 (0.03) u=4 0.46 (0.01) u=9 0.84 (0.00) u=3

-0.02 (0.01) v=14 0.42 (0.02) v=4

0.00 (0.01) u=7 0.45 (0.02) u=7

-0.02 (0.01) v=14 0.42 (0.02) v=4

-0.01 (0.01) v=8

0.02 (0.02) u=9 0.39 (0.01) u=10 0.80 (-) u=1

0.02 (0.01) v=8

-0.02 (0.01) u=7 0.37 (0.02) u=4

0.01 (0.01) v=8

0.0

0.2

0.4

0.6

0.8

1.0

1.0

0.86 (0.00) u=11 0.90 (0.00) u=19 0.91 (0.00) u=19 0.92 (0.00) u=21

0.0

0.2

0.4

0.6

0.8

1.0

-0.02 (0.02) v=8 0.42 (0.02) u=3 0.85 (0.00) u=16 0.89 (0.00) u=23 0.91 (0.00) u=25 0.92 (0.00) u=25

0.45 (0.01) u=5 0.86 (0.00) u=11 0.89 (0.00) u=6 0.91 (0.00) u=6 0.92 (0.01) u=4 0.03 (-) u=1 0.40 (0.02) u=8 0.83 (0.01) u=8 0.89 (0.01) u=2



Note- u and v: the number of equilibrium distributions and flocks in motion in an interval of relocations, respectively. Standard errors are within parantheses. All figures rounded off. ( - ): no standard error calculated because of one replication only.


relocations q s

73


15-03-2012

08:56

Pagina 74

Appendix B: Multi-scale analysis: spatial autoregressive coefficient as a function of the simulation parameters by level of aggregation In empirical applications it is not always evident what the appropriate observation scale is. We therefore estimated the spatial autoregressive coefficient on data that were aggregated at different levels. Particularly, resource and animal densities were averaged for (2 × 2) cells and (4 × 4) cells to render a “new” data set. In the same way as described in the main text, the resulting estimates of ρ were logit transformed and regressed against the simulation parameters. Table S3 shows the parameter estimates. To facilitate comparison the estimates on non-aggregated data in Table 3.2 in the main text are also presented (level of aggregation 1 × 1). ρ Table C.1. 1n( ) as a function of the simulation parameters for the original and aggre1– ρ gated data.

Level of aggregation

CHAPTER 3

2x2

4x4

Perfect Imperfect information information

Perfect Imperfect information information

Constant

1.67 (0.02)

1.49 (0.02)

1.38 (0.02)

1.46 (0.03)

0.43 (0.03)

0.61 (0.04)

s

2.39 (0.03)

2.53 (0.03)

3.39 (0.04)

3.42 (0.04)

3.38 (0.05)

3.24 (0.05)

q

-3.12 (0.03)

-2.40 (0.03)

-2.62 (0.04)

-2.22 (0.04)

-2.12 (0.05)

-1.74 (0.06)

sxq

2.53 (0.05)

1.74 (0.05)

2.12 (0.06)

1.85 (0.07)

1.56 (0.08)

1.76 (0.09)

Animal density

-0.63 (0.00)

-0.56 (0.00)

-0.67 (0.00)

-0.57 (0.00)

-0.58 (0.00)

-0.37 (0.00)

r

0.29 (0.00)

0.27 (0.00)

-0.10 (0.00)

-0.20 (0.00)

-0.22 (0.00)

-0.42 (0.00)

0.86

0.82

0.84

0.77

0.74

0.68

R2

74

1x1 Perfect Imperfect information information

Table C1 shows that the amount of variation in ρ that is explained is high for all the models for both perfect and imperfect information at all levels of aggregation. Furthermore, it shows that the regression coefficients of the variables s, q,animal density and the interaction term s × q have equal signs for all the levels of aggregation. The effect of r, however, is positive for the unaggregated case and turns negative for both the aggregated cases.

15-03-2012 08:56


Proefschrift E. Folmer Pagina 75

75


15-03-2012

08:56

Pagina 76


15-03-2012

4

08:56

Pagina 77

The relative contributions of resource availability and social aggregation to foraging distributions: A spatial lag modelling approach

Eelke O. Folmer and Theunis Piersma

Abstract The spatial distribution of foraging animals simultaneously depends on (1) exogenous environmental variables such as resource availability and abiotic habitat characteristics, and (2) the endogenous variable social aggregation made up of the opposing mechanisms of conspecific attraction and conspecific repulsion. In this paper we develop an exogenous-environment – social aggregation model and use it to analyse the spatial distribution of six abundant shorebird species in the Dutch Wadden Sea at varying resolutions (150 × 150, 200 × 200 and 250 × 250 m). We estimate the model parameters by spatial autoregression. This approach

enables, amongst others, estimation of the direct and indirect effects of an exogenous environmental variable on animal density. The former is given by the regression coefficient and the latter - which is due to the amplification of the direct effect by social aggregation - by the spatial multiplier. At all three levels of resolution and for all species, the explanatory power of social aggregation, measured by Nagelkerke R2, is larger than the contribution of the exogenous environmental variables food availability, silt content, and elevation of the mudflat together. Social aggregation is stronger for dunlin (Calidris alpina), red knot (Calidris canutus) and curlew (Numenius arquata) than for oystercatcher (Haematopus ostralegus), grey plover (Pluvialis squatarola) and bar-tailed godwit (Limosa lapponica). The total impacts (i.e. direct effect plus indirect impacts) of the exogenous environmental predictors tends to substantially exceed the direct effect.

submitted


15-03-2012

08:56

Pagina 78

CHAPTER 4

Introduction

78

Expected intake rate and predation risk are major determinants of foraging distributions (Brown and Kotler 2004, Stephens et al. 2007). Intake rate has been shown to negatively depend on interference behaviour, i.e. interactions such as fighting, stealing prey and monopolization of food patches (Goss-Custard 1980; Sutherland & Koene 1982; Goss-Custard et al. 2001; Vahl et al. 2005). Interference sensitivity is strongly related to attack distance (Stillman et al. 2002), which depends on handling time, which in its turn depends on properties of the predators and their prey (Goss-Custard 1980, Stillman et al. 2002, van Gils and Piersma 2004). Animals may reduce the cost of interference by spacing out (Ens et al. 1990, Stillman et al. 2002, Folmer et al. 2011; Chapter 3, Bijleveld et al. in press; Chapter 5). The basic result of this literature is that if animals are unconstrained in selecting foraging patches, and merely suffer from the co-occurrence of conspecifics, equilibrium spatial distributions arise such that the marginal pay-off amongst patches is equal (Fretwell & Lucas 1969; Kacelnik, Krebs, & Bernstein 1992; Sutherland 1983). The conventional patch selection literature ignores that animals may also benefit from the co-occurrence of conspecifics (Underwood 1982; Krause & Ruxton 2002; Nilsson et al. 2007; Campomizzi et al. 2008). Specifically, the chance of being depredated decreases with group size (Hamilton 1971; Pulliam 1973). Furthermore, the presence of conspecifics provides clues about predation risks (Lima & Dill 1990) and the availability of food (e.g. Camazine et al. 2001; Valone and Templeton 2002; Danchin et al. 2004; Dall et al. 2005; Deygout et al. 2010). In addition, for scrounging individuals the nearby presence of foraging conspecifics may provide foraging opportunities in that prey can be obtained by means of stealing (Giraldeau & Caraco 2000; Rutten et al. 2010). In a review of the literature, Beauchamp (1998) found that for birds intake rate generally increases with group size. We denote the combination of conspecific attraction and repulsion ’social aggregation’ to stress the difference from aggregations resulting from foragers that independently from each other select the same foraging location. The benefits that results from the presence of conspecifics is denoted aggregation economy (Giraldeau and Caraco, 2000). As mudflats are large and open habitats in which the benthic food stocks are buried in the sediment such that the quality of foraging locations can only be learned by trial and error or by close inspection of the mudflat surface, shorebirds foraging on mudflats are ideal to study the effects of the resource distribution and social aggregation on foraging distributions (Piersma et al. 1993, van de Kam et al. 2004, van Gils et al. 2006). To reduce uncertainty in the search process, shorebirds may benefit from information provided by the presence and behaviour of conspecifics (Clark & Mangel 1984; Valone 2007). The average

15-03-2012

08:56

Pagina 79

costs and benefits of conspecific presence, however, vary from species to species. For example, red knots forage on small buried bivalves which they find by remotely sensing hard objects in soft sediments which they repeatedly probe with their bill (Piersma et al. 1998). Once encountered, a prey is retrieved, handled and swallowed intact in seconds (van Gils and Piersma 2004, van de Kam et al. 2004). Because prey processing is so short, kleptoparasitism is not possible and therefore red knots are relatively insensitive to interference (Ens, Esselink, and Zwarts 1990). Therefore knots can pack closely at minor costs (van Gils and Piersma 2004). In contrast, grey plovers locate their polychaete prey visually (Kersten & Piersma 1984). For instance, grey plovers can spot worms moving at the surface of the sediment over distances in the order of 10s of meters. However, even when worms are abundant, the fraction that is visually detectable is usually very low (Zwarts and Wanink 1993). Hence, grey plovers are likely to detect the same prey within distances of 10s of meters from each other and thus may incur interference costs. In addition, they may suffer indirectly from each other’s presence because of prey depression, i.e. worms decrease their surface movements so as not to be detected by predators (Charnov, Orians, & Hyatt 1976; Goss-Custard 1980; Yates, Stillman, & GossCustard 2000). Hence, the presence of conspecifics decreases hunting success over relatively large distances. Thus, for grey plovers interference costs reduce conspecific attraction benefits, and therefore they maintain large inter-individual distances. In a regression for six different species of forager density on food availability and abiotic conditions only, Folmer et al. (2010. Chapter 2) found that the residual variance is substantially larger for red knot than for grey plover. This result is in line with the hypothesis that the former are driven by conspecific attraction and food availability, and the latter mainly by food availability. Although conceptually, its impact on foraging behavior is fairly straightforward, the precise way in which social aggregation should be included in a regression specification is complex. Hence, research on the impacts of resource availability and social aggregation on the spatial distribution of foraging animals has been hampered by the lack of a methodology that allows estimation of their separate effects (Beauchamp 1998; Sumpter 2010). In a previous paper (Folmer et al. 2011), we showed that social aggregation manifests itself as spatial interdependence between neighbouring foraging areas, i.e. an observation (the number of foraging animals) associated with one location depends on the observations (the numbers of foraging animals) at other locations. Furthermore, we suggested to estimate and to test the exogenous-environment - social aggregation model by spatial autoregression (SAR). By means of Monte Carlo simulations we showed that SAR performs well on gridded data. The imposition of a grid of some resolution may, however, lead to the modifiable areal unit problem (MAUP). That is, the chosen grid imposes an

ESTIMATIONS OF FOOD EFFECTS AND SOCIAL AGGREGATION


79


15-03-2012

08:56

Pagina 80

arbitrary measurement system on the spatial process of foraging site selection. As shown by others (e.g. Openshaw 1984; Fotheringham & Wong 1991; Holt et al. 1996; Jelinski & Wu 1996; Heywood, Cornelius, & Carver 1998; Fortin & Dale 2005, Schneider 2009), MAUP can affect parameter estimates in regression analysis. We demonstrated, however, that multi-scale analysis (Wiens 1989) can be applied to obtain robust estimates of the relationship between predictors and response variables. Finally we showed that spatial dependence in the response variable implies that the direct impact of an exogenous environmental predictor like food availability is amplified by the interdependent behaviour of the foraging animals. We demonstrated that SAR allows estimation of the direct effect and the total effect (direct plus all indirect effects), the latter by means of the spatial multiplier. The purpose of the present paper is to estimate the impacts of exogenous predictors and social aggregation on foraging distributions of six abundant shorebird species in the Dutch Wadden Sea at three spatial resolutions using the SAR methodology previously presented by Folmer et al. (2011).

Study area and data collection

CHAPTER 4

The Dutch Wadden Sea The Dutch Wadden Sea is shallow and contains large soft-sediment flats that emerge approximately twice a day during low tide during which they are accessible to shorebirds. Intertidal flats alternate with permanent channels. The flats are characterized by smooth gradients both in terms of abiotic features, such as sediment grain size (Zwarts et al. 2004), and biological properties, such as density of macrozoobenthos (Kraan et al. 2009). The six most abundant wader species are dunlin, red knot, oystercatcher, curlew, grey plover and bar-tailed godwit. The analysis focuses on these species because they are found in sufficiently large numbers for adequate statistical analyses and because there is large variation in flocking patterns between these species.

80

Data collection and preparation As part of a long-term benthic research programme (Piersma et al. 1993; Kraan, van der Meer, et al. 2009), the density of macrozoobenthos was determined in the eastern and western Dutch Wadden Sea in July and September 2004. Benthos sampling was performed over 250 m grids in confined areas at 23 mudflats (sites). For each bird species at each sample station, we determined which prey items were available (not buried too deeply) and ingestible (smaller than maximum length and larger than minimum length) (Zwarts & Wanink 1993). For bivalves we determined the energetic value by measuring the ash free dry mass

15-03-2012

08:56

Pagina 81

(AFDM) in the laboratory (for details see Piersma, de Goeij, et al. 1993, Piersma et al. 1995, Kraan, van Gils et al. 2009). For the specimens that were counted in the field and not brought to the lab (polychaetes and isopods) we obtained estimates of their energetic value from the literature (see Folmer et al. 2010 for details). Maps of the foraging shorebird on the 23 sites were taken from Folmer et al. (2010) and combined with maps of the distributions of species-specific harvestable benthos. The locations of individual birds and flocks could be determined with a precision of approximately 50 m while the benthos data were sampled on a 250 × 250 m grid. Finer resolutions than the 250 × 250 m grid of benthos biomass densities were obtained by thin plate spline interpolation (TPS). The interpolation was obtained by minimization of the residual sum of squares between the data and the predicted surface, constrained by a roughness penalty (Green & Silverman 1993; Wood 2006). The smoothing parameter is automatically chosen by generalized cross validation (GCV). Thin plate spline interpolation is simple, requires no knowledge of spatial model parameters and is suitable for positively skewed data. For red knots a subset of the 23 sites were included in the analysis. The reason is that the population of red knots in the Wadden Sea is highly variable in August because of turnover of two distinct populations. By the beginning of September members of the canutus subspecies have departed while the other subspecies, islandica, has arrived (Zwarts, Blomert & Wanink 1992; Piersma et al. 1993b). For red knot we only considered the 16 sites observed after 1 September. Individual birds were aggregated in grids that fully covered the censused sites (Figure 4.1). The numbers of birds inside the cells were transformed to densities (No × ha-1). The density of a species in each cell was related to the exogenous environmental variables, i.e. density of prey (AFDM × m2), mudflat elevation (m +NAP , the standard Dutch elevation reference) and silt content (% weight) of the sediment (obtained from Zwarts et al. 2004), and to the endogenous variable social aggregation (i.e. the density of birds in neighbouring cells). With respect to the density of prey, we included all relevant benthic species identified as food in the literature that were reasonably abundant (see supplementary material in Folmer et al. 2010 for further information about the benthic species included). Some cells were partially outside the censused site boundaries. They were included in the data set, if at least 50% of the area was inside the site. To account for the disturbance caused by the presence of the observer, cells located near the observation point were removed from the datasets. Depending on species specific sensitivity to observer disturbance, we removed the cells whose centroids were within the following distances from the observer: dunlin

ESTIMATIONS OF FOOD EFFECTS AND SOCIAL AGGREGATION


81


15-03-2012

08:56

Pagina 82

A

censused site individual bird observation point disturbed area

bird density (No./ha) 0.0 – 0.5 0.6 – 1.4 1.5 – 2.7 2.8 – 3.7 3.8 – 4.8 4.9 – 5.6 0

750

1,500 m

B Cerastoderma (g/m2) 0.0 – 0.4 0.5 – 1.9 2.0 – 5.8 5.9 – 10.4 10.5 – 14.3 14.4 – 17.8

C Nereis (g/m2) 0.0 – 0.3 0.4 – 0.9 1.0 – 2.0 2.1 – 3.3 3.4 – 5.4

CHAPTER 4

5.5 – 7.2

82

Figure 4.1. An example of a site with the locations of individual oystercatchers and food resources; resolution: 250 × 250 m. The dots denote individual birds (all three panels). Panel A: Mean bird density (No × ha-1); Panel B: Cockle (Cerastoderma edule) biomass (ash free dry mass) (g × m-2); Panel C: Biomass (AFDM) of the polychaete Nereis diversicolor (g × m-2). The “hole” in the middle is the disturbed area around the observer. Cells with centroid within the disturbed area are removed from the dataset. For further details see Methods.


15-03-2012

08:56

Pagina 83

and red knot: 150 m, oystercatcher, grey plover and bar-tailed godwit: 200 m, curlew: 300 m (Spaans, Bruinzeel, & Smit 1996). The resulting lattices contained all relevant information for statistical analysis, i.e. bird and prey densities, the abiotic habitat characteristics and the geographical coordinates of each cell. Each site was divided into cells of 250 × 250 m, 200 × 200 m and 150 × 150 m, respectively. The data sets consisted of the aggregate of the cells over the sites. The total number of observations (which varies by cell size) for each species is given in Table 4.1. To check the robustness of the results, we estimated models for the 250 × 250 m, 200 × 200 m, and 150 × 150 m resolutions.

Statistical analysis The spatial lag model We estimated the exogenous-environment – social aggregation model by means of the spatial lag model which is made up of two systematic components, i.e. the spatial autoregressive component representing social aggregation and the set of exogenous variables representing the exogenous environment. The spatial lag model (Anselin 1988, Haining 2003, LeSage and Pace 2009) (in matrix notation) reads:

where y is an n × 1 vector of observations on the dependent variable (in the present case bird density), X is an n × k data matrix of explanatory variables with associated coefficient vector β, ε is an n × 1 vector of error terms which follows a normal distribution, i.e. ε ~ N(0, σ 2 1n). W is the n × n spatial weights matrix and ρ the spatial autoregression coefficient or spatial lag parameter. The spatial weights matrix W represents spatial dependence (or connectivity) among the observations. Various types of W matrices may be employed (see Fortin and Dale 2005). We defined cells as spatially dependent, if the distance between their centroids was less than or equal to 750 m. The limit of 750 m is based on the assumption that it is roughly the maximum distance over which the benefits of conspecific attraction extend. Spatial dependence was measured by inverse distance. That is, Wij = 1/dij if the distance between the centroids of cell i and j